Differential Equations

The given equation is:

$ x \frac{d y}{d x}-y=x^2 \cot x $

To get it into the standard form $\frac{d y}{d x}+P(x) y=Q(x)$, we divide the entire equation by $x$ :

$ \frac{d y}{d x}-\frac{1}{x} y=x \cot x $

Here, $P(x)=-\frac{1}{x}$ and $Q(x)=x \cot x$.

Finding the Integrating Factor (I.F.)

The integrating factor is calculated as:

$ \text { I.F. }=e^{\int P(x) d x}=e^{\int-\frac{1}{x} d x}=e^{-\ln x}=e^{\ln \left(x^{-1}\right)}=\frac{1}{x} $

The solution to the differential equation is:

$ \begin{gathered} y \cdot(I . F .)=\int Q(x) \cdot(I . F .) d x \\ \frac{y}{x}=\int(x \cot x) \cdot \frac{1}{x} d x \\ \frac{y}{x}=\int \cot x d x \\ \frac{y}{x}=\ln (\sin x)+C \\ y=x \ln (\sin x)+C x \end{gathered} $

Applying the Initial Condition

We are given $y\left(\frac{\pi}{2}\right)=\frac{\pi}{2}$. Plugging these values in:

$ \frac{\pi}{2}=\frac{\pi}{2} \ln \left(\sin \frac{\pi}{2}\right)+C\left(\frac{\pi}{2}\right) $

So, the particular solution is $y=x \ln (\sin x)+x$.

We need to find the value of $6 y\left(\frac{\pi}{6}\right)-8 y\left(\frac{\pi}{4}\right)$.

• Calculate $y\left(\frac{\pi}{6}\right)$ :

$ y\left(\frac{\pi}{6}\right)=\frac{\pi}{6} \ln \left(\sin \frac{\pi}{6}\right)+\frac{\pi}{6}=\frac{\pi}{6} \ln \left(\frac{1}{2}\right)+\frac{\pi}{6}=\frac{\pi}{6}(1-\ln 2) $

• Calculate $y\left(\frac{\pi}{4}\right)$ :

$ y\left(\frac{\pi}{4}\right)=\frac{\pi}{4} \ln \left(\sin \frac{\pi}{4}\right)+\frac{\pi}{4}=\frac{\pi}{4} \ln \left(\frac{1}{\sqrt{2}}\right)+\frac{\pi}{4}=\frac{\pi}{4} \ln \left(2^{-1 / 2}\right)+\frac{\pi}{4}=\frac{\pi}{4}\left(1-\frac{1}{2} \ln 2\right) $

• Calculate $y\left(\frac{\pi}{4}\right)$ :

$ y\left(\frac{\pi}{4}\right)=\frac{\pi}{4} \ln \left(\sin \frac{\pi}{4}\right)+\frac{\pi}{4}=\frac{\pi}{4} \ln \left(\frac{1}{\sqrt{2}}\right)+\frac{\pi}{4}=\frac{\pi}{4} \ln \left(2^{-1 / 2}\right)+\frac{\pi}{4}=\frac{\pi}{4}\left(1-\frac{1}{2} \ln 2\right) $

Now, substitute these into the expression:

$ \begin{aligned} 6 y\left(\frac{\pi}{6}\right)-8 y\left(\frac{\pi}{4}\right) & =6\left[\frac{\pi}{6}(1-\ln 2)\right]-8\left[\frac{\pi}{4}\left(1-\frac{1}{2} \ln 2\right)\right]=\pi(1-\ln 2)-2 \pi\left(1-\frac{1}{2} \ln 2\right) \\ = & \pi-\pi \ln 2-2 \pi+\pi \ln 2=-\pi \end{aligned} $

Given equation is :

$ x \frac{d y}{d x}-\sin 2 y=x^3\left(2-x^3\right) \cos ^2 y, \quad x \neq 0 $

Divide by $x \cos ^2 y$

$ \sec ^2 y \frac{d y}{d x}-\frac{\sin 2 y}{x \cos ^2 y}=x^2\left(2-x^3\right) $

$ \sec ^2 y \frac{d y}{d x}-\frac{2 \tan y}{x}=2 x^2-x^5 $

Substitute $\tan y=t$ and differentiate w.r.t $x$

$ \sec ^2 y \frac{d y}{d x}=\frac{d t}{d x} $

Equation (1) becomes linear differential equation.

$\frac{d t}{d x}-\frac{2}{x} t=2 x^2-x^5$

Integrating factor $I F=e^{\int\left(-\frac{2}{x}\right) d x}=e^{-2 \ln x}=e^{\ln \left(\frac{1}{x^2}\right)}=\frac{1}{x^2}$

Solution of linear differential equation :

$ t \cdot I F=\int(I F)\left(2 x^2-x^5\right) d x $

$\Rightarrow $ $t \cdot \frac{1}{x^2}=\int \frac{1}{x^2}\left(2 x^2-x^5\right) d x$

$\Rightarrow $ $ \frac{t}{x^2}=\int\left(2-x^3\right) d x=2 x-\frac{x^4}{4}+C $

put value of $t=\tan y$

$ \frac{\tan y}{x^2}=2 x-\frac{x^4}{4}+C $

using condition $y(2)=0$ to find $C$.

$ \frac{\tan (0)}{2^2}=2 \times 2-\frac{2^4}{4}+C \Longrightarrow C=0 $

So, solution of differential equation is

$ \tan y=2 x^3-\frac{x^6}{4} $

to calculate $\tan (y(1))$, put $x=1$ in the solution.

$ \tan (y(1))=2(1)^3-\frac{(1)^6}{4} $

$=2-\frac{1}{4}=\frac{7}{4}$

$ \begin{aligned} & x^4 d y+\left(4 x^3 y+2 \sin x\right) d x=0 \\ & \frac{d y}{d x}+\frac{4 x^3}{x^4} y=\frac{-2 \sin x}{x^4} \\ & \frac{d y}{d x}+\frac{4}{x} y=\frac{-2 \sin x}{x^4} \\ & \text { IF }=e^{\int \frac{4}{x} d x=e^{\ln x^4}=x^4} \\ & x^4 \frac{d y}{d x}+4 x^3 y=-2 \sin x \\ & \frac{d}{d x}\left(x^4 y\right)=-2 \sin x \\ & \Rightarrow x^4 y=2 \cos x+c \\ & y\left(\frac{\pi}{2}\right)=0 \\ & \Rightarrow c=0 \\ & \therefore x^4 y=\cos x \\ & \Rightarrow y=\frac{2 \cos x}{x^4} \end{aligned} $

$ \begin{aligned} & y\left(\frac{\pi}{3}\right)=\frac{2 \cos \left(\frac{\pi}{3}\right)}{\left(\frac{\pi}{3}\right)^4}=\frac{1}{\left(\frac{\pi}{3}\right)^4}=\frac{81}{\pi^4} \\ & \pi^4 y\left(\frac{\pi}{3}\right)=81 \end{aligned} $

$ \begin{aligned} &\begin{aligned} & \int \frac{\cos y}{1+2 \sin y} d y=\int \frac{1}{16(\sqrt{x+9 \sqrt{x}})(4+\sqrt{9+\sqrt{x}})} d x \\ & \Rightarrow \frac{1}{2} \int \frac{2 \cos y}{1+2 \sin y} d y=\int \frac{1}{4 t} d t \end{aligned}\\ &\text { Put } 4+\sqrt{9+\sqrt{x}}=t\\ &\begin{aligned} & \Rightarrow \frac{1}{2 \sqrt{9+\sqrt{x}}} \cdot \frac{1}{2 \sqrt{x}} d x=d t \Rightarrow \frac{1}{4 \sqrt{x+9 \sqrt{x}}} d x=d t \\ & \Rightarrow \frac{1}{2} \log |1+2 \sin y|=\frac{1}{4} \log |t|+\log c \Rightarrow \sqrt{1+2 \sin y}=c t^{1 / 4} \Rightarrow \frac{1}{4} \log |t|+\log c \\ & \Rightarrow \sqrt{1+2 \sin y}=c t^{1 / 4} \Rightarrow \sqrt{1+2 \sin y}=c(4+\sqrt{9+\sqrt{x}})^{1 / 4} \\ & \left(256, \frac{\pi}{2}\right) \Rightarrow \sqrt{3}=c(\sqrt{3}) \Rightarrow c=1 \\ & (49, \alpha) \Rightarrow \sqrt{1+2 \sin \alpha}=(1) 8^{1 / 4} \\ & \Rightarrow 1+2 \sin \alpha=2 \sqrt{2} \Rightarrow 2 \sin \alpha=2 \sqrt{2}-1 \end{aligned} \end{aligned} $

$ \begin{aligned} & x d y-y d x=\sqrt{x^2+y^2} d x \\ & \frac{x d y-y d x}{x^2}=\frac{\sqrt{x^2+y^2}}{x^2} d x \end{aligned} $

$ \begin{aligned} & \int \frac{d(y / x)}{\sqrt{1+(y / x)^2}}=\int \frac{1}{x} d x \\ & \log \left(\frac{y}{x}+\sqrt{1+\left(\frac{y}{x}\right)^2}\right)=\log x+\log C \\ & y+\sqrt{x^2+y^2}=x^2 \\ & y-\sqrt{x^2+y^2}=-1 \\ & 2 y=x^2-1 \\ & y(3)=4 \end{aligned} $

$ \begin{aligned} & \frac{d y}{d x}-2 \cos x \cdot y=\cos x(2+3 \sin x) \\ & I \cdot F=e^{-2 \sin x} \\ & G \cdot S y \cdot e^{-2 \sin x}=\int e^{-2 \sin x} \cos x(2+3 \sin x) d x \\ & \text { Put } \begin{array}{l} \sin x=t \\ \cos x \cdot d x=d t \end{array} \end{aligned} $

$ y=e^{-2 \sin x}\left(\frac{2+3 \sin x}{-2}-\frac{3}{4}\right)+c $

When $x=0, c=0$

When $x=\frac{\pi}{6}, y=-\frac{5}{2}$

Given

$ (1+x^2)\,dy+(y-\tan^{-1}x)\,dx=0,\qquad y(0)=1, $

divide by $dx$:

$ (1+x^2)\frac{dy}{dx}+y-\tan^{-1}x=0 $

$ \frac{dy}{dx}+\frac{1}{1+x^2}y=\frac{\tan^{-1}x}{1+x^2}. $

This is linear. Integrating factor:

$ \mu(x)=\exp\!\left(\int \frac{1}{1+x^2}\,dx\right)=e^{\tan^{-1}x}. $

So

$ \frac{d}{dx}\left( y e^{\tan^{-1}x}\right)=e^{\tan^{-1}x}\cdot \frac{\tan^{-1}x}{1+x^2}. $

Let $t=\tan^{-1}x$, then $dt=\frac{dx}{1+x^2}$. Hence

$ y e^t=\int t e^t\,dt + C = e^t(t-1)+C. $

Therefore

$ y=t-1+Ce^{-t}=\tan^{-1}x-1+Ce^{-\tan^{-1}x}. $

Use $y(0)=1$ and $\tan^{-1}0=0$:

$ 1=0-1+C \implies C=2. $

So

$ y(1)=\frac{\pi}{4}-1+2e^{-\pi/4}=\frac{2}{e^{\pi/4}}+\frac{\pi}{4}-1. $

Correct option: B.

$\begin{aligned} & f(x)=x-1 \\ & f(f(x))=f(x)-1=x-1-1=x-2 \\ & g(f(f(x)))=e^{x-2} \\ & \therefore \frac{d y}{d x}=e^{-2 \sqrt{x}} \times e^{x-2}-\frac{1}{\sqrt{x}} y \\ & \frac{d y}{d x}+\frac{1}{\sqrt{x}} y=e^{x-2 \sqrt{x}-2} \text { which is L.D.E } \\ & \text { I.F. }=e^{\int \frac{d y}{\sqrt{x}}}=e^{2 \sqrt{x}} \end{aligned}$

Its solution is

$\begin{aligned} & y \times e^{2 \sqrt{x}}=\int e^{2 \sqrt{x}} \times e^{x-2 \sqrt{x}-2} d x+c \\ & y \times e^{2 \sqrt{x}}=\int e^{x-2} d x+c \\ & y \times e^{2 \sqrt{x}}=e^{x-2}+c \end{aligned}$

Given $\mathrm{x}=0, \mathrm{y}=0 \Rightarrow 0=\mathrm{e}^{-2}+\mathrm{c} \quad ; \mathrm{c}=-\mathrm{e}^{-2}$

$\therefore \mathrm{y} \times \mathrm{e}^{2 \sqrt{x}}=\mathrm{e}^{\mathrm{x}-2}-\mathrm{e}^{-2}$

when $\mathrm{x}=1, \mathrm{y} \times \mathrm{e}^2=\mathrm{e}^{-1}-\mathrm{e}^{-2}$

$y=\frac{e^{-1}-e^{-2}}{e^2}=\frac{\frac{1}{e}-\frac{1}{e^2}}{e^2}=\frac{e^2-e}{e^5}=\frac{e-1}{e^4}$

Option (1) is correct

$\begin{aligned} &\begin{aligned} & \left(x^2+1\right) \frac{d y}{d x}-2 x y=\left(x^4+2 x^2+1\right) \cos x \\ & \frac{d y}{d x}-\left(\frac{2 x}{x^2+1}\right) y=\frac{\left(x^2+1\right)^2 \cos x}{c x^2+1}=\left(x^2+1\right) \cos x \end{aligned}\\ &\text { (Linear D.E) }\\ &\mathrm{P}=\frac{-2 \mathrm{x}}{\mathrm{x}^2+1}, \mathrm{Q}=\left(\mathrm{x}^2+1\right) \cos \mathrm{x} \end{aligned}$

$\begin{aligned} & \text { I.F }=\mathrm{e}^{\int P d x}=\mathrm{e}^{\int \frac{-2 x}{x^2+1} d x}=\frac{1}{x^2+1} \\ & y \cdot \frac{1}{x^2+1}=\int\left(x^2+1\right) \cos x \cdot \frac{1}{x^2+1} d x \\ & \frac{y}{x^2+1}=\sin x+c \Rightarrow y \cos =1 \Rightarrow c=1 \end{aligned}$

$\begin{aligned} & y=\left(x^2+1\right)(\sin x+1) \\ & \int_{-3}^3 y d x=\int_{-3}^3\left(x^2+1\right)(\sin x+1) \\ & d x=\int_{-3}^3 x^2 \sin x+x^2 \sin x+1 d x \\ & \Rightarrow \int_{-3}^3 x^2 \sin x d x+\int_{-3}^3 x^2 d x+\int_{-3}^3 \sin x d x+\int_{-3}^3 1 d x \\ & =0+18+0+6=24 \end{aligned}$

$\begin{aligned} & x\left(x^2+e^x\right) d y+\left(e^x(x-2) y-x^3\right) d x=0 \\ & \frac{d y}{d x}+\frac{e^x(x-2)}{x\left(x^2+e^x\right)} y=\frac{x^3}{x\left(x^2+e^x\right)} \\ & \text { I.F. }=e^{\iint \frac{e^x(x-2)}{x\left(x^2+e^x\right)} d x} \\ & =e^{\int \frac{e^x+2 x}{e^x+x^2} d x-\int \frac{2}{x} d x} \\ & =e^{\ln e^{-x}+x^2-2 \ln x} \end{aligned}$

$\begin{aligned} & =\frac{e^x+x^2}{x^2} \\ & \therefore \quad y\left(\frac{e^2+x^2}{x^2}\right)=\int d x+c \\ & \Rightarrow \quad y\left(\frac{e^x+x^2}{x^2}\right)=x+c \end{aligned}$

Also, $y(1)=0$

$\begin{aligned} & \Rightarrow \quad c=-1 \\ & \therefore \quad y\left(\frac{e^2+x^2}{x^2}\right)=x-1 \end{aligned}$

Hence, $y(2)=\frac{4}{e^2+4}$

$\begin{aligned} & \left(7 x^4 \cot y-e^x \operatorname{cosec} y\right) \frac{d x}{d y}=x^5 \\ & x^5 \frac{d y}{d x}-7 x^4 \cot y=-e^x \operatorname{cosec} y \\ & \frac{d y}{d x}-\frac{7}{x} \cot y=-\frac{e^x}{x^5} \operatorname{cosec} y \\ & \sin y \frac{d y}{d x}-\frac{7}{x} \cos y=-\frac{e^x}{x^5} \\ & \text { Let }-\cos y=t \\ & \sin y \frac{d y}{d x}=\frac{d t}{d x} \\ & \therefore \frac{d t}{d x}+\frac{7}{x} t=-\frac{e^x}{x^5} \end{aligned}$

$\begin{aligned} & \therefore \text { I.F. }=e^{\int \frac{7}{x} d x}=x^7 \\ & t \cdot x^7=\int \frac{-e^x}{x^5} \cdot x^7 d x \\ & -\cos y \cdot x^7=-\int e^x x^2 d x \\ & \cos y x^7=e^x\left(x^2-2 x+2\right)+c \\ & \because \quad x=1 \text { then } y=\frac{\pi}{2} \Rightarrow c=-e \\ & \therefore \quad \cos y \cdot x^7=e^x\left(x^2-2 x+2\right)-e \\ & \text { When } x=2 \text { then } \cos y=\frac{2 e^2-e}{128} \end{aligned}$

$\begin{aligned} & \frac{d y}{d x}+3\left(\tan ^2 x\right) y+3 y=\sec ^2 x \\ & \Rightarrow \frac{d y}{d x}+3 \sec ^2 x y=\sec ^2 x \\ & \text { I.F }=e^{\int 3 \sec ^2 x d x} \\ & \quad=e^{3 \tan x} \end{aligned}$

$\begin{aligned} & y \cdot e^{\tan x}=\int e^{3 \tan x} \cdot \sec ^2 x d x+c \\ & y \cdot e^{3 \tan x}=\frac{e^{3 \tan x}}{3}+c \\ & \text { Also } f(0)=\frac{1}{3}+e^3 \\ & \Rightarrow\left(\frac{1}{3}+e^3\right)=\frac{1}{3}+c \\ & \Rightarrow c=e^3 \\ & \therefore y \cdot e^{3 \tan x}=\frac{e^{3 \tan x}}{3}+e^3 \\ & \text { Put } x=\frac{\pi}{4} \end{aligned}$

$y e^3=\frac{e^3}{3}+e^3 \Rightarrow y=\frac{4}{3}$

To solve the given problem, let's start by considering the equation:

$ \int_0^x g(t) \, dt = x - \int_0^x \tan g(t) \, dt $

Differentiate both sides with respect to $ x $:

$ g(x) = 1 - xg(x) $

Rearranging gives:

$ g(x)(1 + x) = 1 \implies g(x) = \frac{1}{1 + x} $

Now, consider the differential equation:

$ \frac{dy}{dx} - y \tan x = 2(x+1) \sec x g(x) $

Substitute $ g(x) = \frac{1}{1 + x} $:

$ \frac{dy}{dx} - y \tan x = 2(x+1) \sec x \cdot \frac{1}{1 + x} $

This simplifies to:

$ \frac{dy}{dx} - y \tan x = 2 \sec x $

To solve this, we use an Integrating Factor (I.F):

$ \text{I.F} = e^{\int \tan x \, dx} = e^{-\ln \cos x} = \cos x $

Multiply the entire differential equation by the Integrating Factor:

$ \cos x \cdot \frac{dy}{dx} - y \cdot \sin x = 2 $

This implies:

$ \frac{d}{dx}(y \cos x) = 2 $

Integrate both sides with respect to $ x $:

$ y \cos x = \int 2 \, dx = 2x + c $

Given the initial condition $ y(0) = 0 $:

$ 0 \cdot 1 = 2 \cdot 0 + c \implies c = 0 $

Thus:

$ y \cos x = 2x $

Evaluate at $ x = \frac{\pi}{3} $:

$ y \cdot \frac{1}{2} = 2 \cdot \frac{\pi}{3} $

Solving for $ y $:

$ y = \frac{4 \pi}{3} $

Therefore, $ y\left(\frac{\pi}{3}\right) = \frac{4 \pi}{3} $.

$\begin{aligned} & \text { If } \mathrm{e}^{\int \tan x d x}=\mathrm{e}^{\ln (\sec x)}=\sec x \\ & \therefore y \cdot \sec x=\int\left\{\frac{2+\sec x}{(1+2 \sec x)^2}\right\} \sec x d x \end{aligned}$

$\begin{aligned} &\begin{aligned} & =\int \frac{2 \cos x+1}{(\cos x+2)^2} d x \text { Let } \cos x=\frac{1-t^2}{1+t^2} \\ & =\int \frac{2\left(\frac{1-t^2}{1+t^2}\right)+1}{\left(\frac{1-t^2}{1+t^2}+2\right)^2} 2 d t \\ & =\int \frac{2-2 t^2+1+t^2}{\left(1-t^2+2+2 t^2\right)^2} \times 2 d t \\ & =2 \int \frac{3-t^2}{\left(t^2+3\right)^2} d t \end{aligned}\\ &\text { Let } \mathrm{t}+\frac{3}{\mathrm{t}}=\mathrm{u}\\ &\left(1-\frac{3}{\mathrm{t}^2}\right) \mathrm{dt}=\mathrm{du} \end{aligned}$

$=-2 \int \frac{\mathrm{du}}{\mathrm{u}^2}$

$\begin{aligned} & y \cdot(\sec x)=\frac{2}{u}+c \\ & y \cdot \sec x=\frac{2}{t+\frac{3}{t}}+c\quad\text{..... (I)} \end{aligned}$

$\begin{aligned} & \text { At } \mathrm{x}=\frac{\pi}{3}, \mathrm{t}=\tan \frac{\mathrm{x}}{2}=\frac{1}{\sqrt{3}} \\ & \text { 2. } \frac{\sqrt{3}}{10}=\frac{2}{\frac{1}{\sqrt{3}}+3 \sqrt{3}}+\mathrm{c} \\ & \text { 2. } \frac{\sqrt{3}}{10}=\frac{2 \sqrt{3}}{10}+\mathrm{c} \Rightarrow \mathrm{C}=0 \\ & \text { At } \mathrm{x}=\frac{\pi}{4}, \mathrm{t}=\tan \frac{\mathrm{x}}{2}=\sqrt{2}-1 \end{aligned}$

$\begin{aligned} & \therefore y \cdot \sqrt{2}=\frac{2}{\sqrt{2}-1+\frac{3}{\sqrt{2}-1}} \\ & y \cdot \sqrt{2}=\frac{2(\sqrt{2}-1)}{6-2 \sqrt{2}} \\ & y=\frac{\sqrt{2}(\sqrt{2}-1)}{2(3-\sqrt{2})}=\frac{1}{\sqrt{2}} \times \frac{2 \sqrt{2}-1}{7} \\ & =\frac{4-\sqrt{2}}{14} \end{aligned}$

$\begin{aligned} & \cos x(\ln (\cos x))^2 d y+(\sin x-3 y(\sin x) \ln (\cos x)) d x=0 \\ & \cos x(\ln (\cos x))^2 \frac{d y}{d x}-3 \sin x \cdot \ln (\cos x) y=-\sin x \\ & \frac{d y}{d x}-\frac{3 \tan x}{\ln (\cos x)} y=\frac{-\tan x}{(\ln (\cos x))^2} \\ & \frac{d y}{d x}+\frac{3 \tan x}{\ln (\sec x)} y=\frac{-\tan x}{(\ln (\sec x))^2} \\ & \text { I.F. }=e^{\int \frac{3 \tan x}{\ln (\sec x)} d x}=(\ln (\sec x))^3 \end{aligned}$

$\begin{aligned} & y \times(\ln (\sec x))^3=-\int \frac{\tan x}{(\ln (\sec x))^2}(\ln (\sec x))^3 d x \\ & y \times(\ln (\sec x))^3=-\frac{1}{2}(\ln (\sec x))^2+C \\ & \text { Given }: x=\frac{\pi}{4}, y=-\frac{1}{\ln 2} \\ & \frac{-1}{\ln 2} \times(\ln \sqrt{2})^3=-\frac{1}{2} \times(\ln \sqrt{2})^2+C \\ & \Rightarrow \frac{-1}{8 \ln 2} \times(\ln 2)^3=\frac{-1}{2} \times \frac{1}{4}(\ln 2)^2+C \\ & -\frac{1}{8}(\ln 2)^2=\frac{-1}{8}(\ln 2)^2+C \\ & \Rightarrow C=0 \\ & \therefore y(\ln (\sec x))^3=\frac{-1}{2}(\ln (\sec x))^2+0 \\ & y=\frac{-1}{2 \ln (\sec x)} \\ & y=\frac{1}{2 \ln (\cos x)} \end{aligned}$

$\begin{aligned} & \therefore y\left(\frac{\pi}{6}\right)=\frac{1}{2 \ln \left(\cos \frac{\pi}{6}\right)} \\ & =\frac{1}{2 \ln \left(\frac{\sqrt{3}}{2}\right)} \\ & =\frac{1}{2\left(\frac{1}{2} \ln 3-\ln 2\right)} \\ & =\frac{1}{\ln 3-\ln 4} \end{aligned}$

$\begin{aligned} &\int_0^{\mathrm{x}} \mathrm{tf}(\mathrm{t}) \mathrm{dt}=\mathrm{x}^2+(\mathrm{x}), \mathrm{x}>0\\ &\text { Diff. both side w.r. to } x\\ &\begin{aligned} & x f(x)=x^2 f^{\prime}(x)+2 x f(x) \\ & -x f(x)=x^2 f^{\prime}(x) \\ & \int \frac{f^{\prime}(x)}{f(x)} d x=\int \frac{-1}{x} d x \\ & \operatorname{logdf}(x)=-\log x+\log c \\ & f(x)=\frac{c}{x} \end{aligned}\\ &\mathrm{f}(2)=3 \Rightarrow 3=\frac{\mathrm{c}}{2} \Rightarrow \mathrm{c}=6\\ &f(x)=\frac{6}{x}\\ &f(6)=1\quad\therefore (1) \end{aligned}$

$\begin{aligned} & \left(1+x^2\right) \frac{d y}{d x}+x y=5 x^1 \sqrt{1+x^2} \\ & \frac{d y}{d x}+\frac{x y}{1+x^2}=\frac{5 x^2}{\sqrt{1+x^2}} \\ & \therefore \text { I.F. }=e^{\int \frac{x}{1+x^2} d x}=e^{\frac{\ln \left(1+x^2\right)}{2}}=\sqrt{1+x^2} \\ & \therefore y \sqrt{1+x^2}=\int \frac{5 x^2}{\sqrt{1+x^2}} \cdot \sqrt{1+x^2} d x \\ & \therefore y \sqrt{1+x^2}=\int \frac{5 x^2}{\sqrt{1+x^2}} \cdot \sqrt{1+x^2} d x \\ & y \sqrt{1+x^2}=\frac{5 x^3}{3}+C \\ & \because y(0)=0 \Rightarrow 0=0+C \Rightarrow C=0 \\ & \therefore y=\frac{5 x^3}{3 \sqrt{1+x^2}} \\ & y(\sqrt{3})=\frac{15 \sqrt{3}}{32}=\frac{5 \sqrt{3}}{2} \end{aligned}$

$\begin{aligned} \quad y d y & =(x d y-y d x) \sin \left(\frac{x}{y}\right) \\ \frac{d y}{y} & =\left(\frac{x d y-y d x}{y^2}\right) \sin \left(\frac{x}{y}\right) \\ \frac{d y}{y} & =\sin \left(\frac{x}{y}\right) d\left(-\frac{x}{y}\right) \\ \Rightarrow \quad \ell n y & =\cos \frac{x}{y}+C \end{aligned}$

$\begin{aligned} & x(1)=\frac{\pi}{2} \Rightarrow 0=\cos \frac{\pi}{2}+C \Rightarrow C=0 \\ & \text { थny }=\cos \frac{x}{y} \\ & \text { but } y=2 \Rightarrow \cos \frac{x}{2}=\ln 2 \\ & \qquad \begin{aligned} \cos x & =2 \cos ^2 \frac{x}{2}-1 \\ & =2(\ln 2)^2-1 \end{aligned} \end{aligned}$

$\begin{aligned} & \frac{d y}{d x}=\frac{2(3+y) \cdot e^{2 x}}{7+e^{2 x}} \\ & \frac{d y}{d x}-\frac{2 y \cdot e^{2 x}}{7+e^{2 x}}=\frac{6 \cdot e^{2 x}}{7+e^{2 x}} \\ & \text { I.F. }=e^{-\int \frac{2 e^{2 x}}{7+e^{2 x}}}=\frac{1}{7+e^{2 x}} \end{aligned}$

$\begin{aligned} & \therefore y \cdot \frac{1}{7+\mathrm{e}^{2 \mathrm{x}}}=\int \frac{6 \mathrm{e}^{2 \mathrm{x}}}{\left(7+3^{2 x}\right)^2} \mathrm{dx} \\ & \quad \frac{\mathrm{y}}{7+\mathrm{e}^{2 \mathrm{x}}}=\frac{-3}{7+\mathrm{e}^{2 \mathrm{x}}}+C \\ & (0,5) \Rightarrow \frac{5}{8}=\frac{-3}{8}+C \Rightarrow C=1 \\ & \therefore y=-3+7+\mathrm{e}^{2 \mathrm{x}} \\ & y=\mathrm{e}^{2 \mathrm{x}}+4 \\ & \therefore \mathrm{k}=8 \end{aligned}$

$\begin{aligned} & \frac{d x}{d y}+\frac{x}{1+y^2}=\frac{2 e^{\tan ^{-1} y}}{1+y^2} \\ & \text { I.F. }=e^{\tan ^{-1} y} \\ & x^{\tan ^{-1} y}=\int \frac{2\left(e^{\tan ^{-1} y}\right)^2 d y}{1+y^2} \\ & \text { Put } \tan ^{-1} y=t, \frac{d y}{1+y^2}=d t \\ & x e^{\tan ^{-1} y}=\int 2 e^{2 t} d t \\ & x e^{\tan ^{-1} y}=e^{2 \tan ^{-1} y}+c \\ & x=e^{\tan ^{-1} y}+c e^{-\tan ^{-1} y} \\ & \because y=0, x=1 \\ & 1=1+c \Rightarrow c=0 \\ & y=\frac{1}{\sqrt{3}}, x=e^{\pi / 6} \end{aligned}$

$\begin{aligned} & y^2 d x+\left(x-\frac{1}{y}\right) d y=0 \\ & y^2 d x=\left(\frac{1}{y}-x\right) d y \\ & \Rightarrow y^2 \frac{d x}{d y}=\frac{1}{y}-x \\ & \Rightarrow \frac{d x}{d y}+\frac{x}{y^2}=\frac{1}{y^3} \\ & \text { I.F. }=e^{\int \frac{1}{y^2} d y}=e^{\frac{-1}{y}}\end{aligned}$

$\therefore$ Solution is

$ x e^{\frac{-1}{y}}=\int e^{-\frac{1}{y}} \times \frac{1}{y^3} d y+C $

Let $\frac{-1}{y}=t$

$ \Rightarrow \frac{1}{y^2} d y=d t $

$\begin{aligned} & \Rightarrow x e^{-\frac{1}{y}}=-\int e^t t d t+C \\ & \Rightarrow x e^{-\frac{1}{y}}=-e^t(t-1)+C \\ & \Rightarrow x e^{-\frac{1}{y}}=-e^{\frac{-1}{y}}\left(\frac{-1}{y}-1\right)+C\end{aligned}$

$\begin{aligned} & f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(x) \\ & \text { Put }=x=y=0 \\ & f(0)=f(0) f^{\prime}(0)+f^{\prime}(0) f(0) \\ & f^{\prime}(0)=\frac{1}{2} \\ & \text { Put } y=0 \\ & f(x)=f(x) f^{\prime}(0)+f^{\prime}(x) f(0) \\ & f(x)=\frac{1}{2} f(x)+f^{\prime}(x) \\ & f^{\prime}(x)=\frac{f(x)}{2}\end{aligned}$

$ \frac{d y}{d x}=\frac{y}{2} \Rightarrow \int \frac{d y}{y}=\int \frac{d x}{2} $

$ \Rightarrow \ln y=\frac{x}{2}+c $

$\begin{aligned} & \because \mathrm{f}(0)=1 \Rightarrow \mathrm{C}=0 \\ & \ell \mathrm{ny}=\frac{\pi}{2} \Rightarrow \mathrm{f}(\mathrm{x})=\mathrm{e}^{\mathrm{x} / 2} \\ & \ln \mathrm{f}(\mathrm{n})=\frac{\mathrm{n}}{2} \\ & \sum_{\mathrm{n}=1}^{100} \ell \mathrm{f}(\mathrm{n})=\frac{1}{2} \sum_{\mathrm{n}=1}^{100} \mathrm{n}=\frac{5050}{2} \\ & =2525\end{aligned}$

$ \begin{aligned} & \mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x}) \mathrm{f}(\mathrm{y}) \\ & \Rightarrow \mathrm{f}(\mathrm{x})=\mathrm{e}^{\lambda \mathrm{x}} \mathrm{f}^{\prime}(0)=4 \mathrm{a} \\ & \Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=\lambda \mathrm{e}^{\lambda \mathrm{x}} \Rightarrow \lambda=4 \mathrm{a} \end{aligned} $

So, $\mathrm{f}(\mathrm{x})=\mathrm{e}^{4 \mathrm{ax}}$

$ \begin{aligned} & \mathrm{f}^{\prime \prime}(\mathrm{x})-3 \mathrm{af}^{\prime}(\mathrm{x})-\mathrm{f}(\mathrm{x})=0 \\ & \Rightarrow \lambda^2-3 \mathrm{a} \lambda-1=0 \\ & \Rightarrow 16 \mathrm{a}^2-12 \mathrm{a}^2-1=0 \Rightarrow 4 \mathrm{a}^2=1 \Rightarrow \mathrm{a}=\frac{1}{2} \end{aligned} $

$\int_\limits0^x \sqrt{1-\left(y^{\prime}(t)\right)^2} d t=\int_\limits0^x y(t) d t$

Differentiating both side

$\begin{aligned} & \sqrt{1-\left(y^{\prime}(x)\right)^2}=y(x) \\ & \left(\frac{d y}{d x}\right)^2+y^2=1 \\ & y^{\prime 2}+y^2=1 \\ & 2 y^{\prime} y^{\prime \prime}+2 y y^{\prime}=0 \\ & y^{\prime \prime}+y=0 \end{aligned}$

$\therefore$

$\begin{aligned} & \left(x^2+y^2\right) d x-5 x y d y=0 \\ & \frac{d y}{d x}=\frac{x^2+y^2}{5 x y} \\ & \text { Let } y=v x \\ & \frac{d y}{d x}=v+x \frac{d v}{d x} \\ & V+x \frac{d v}{d x}=\frac{1+v^2}{5 v} \\ & x \frac{d v}{d x}=\frac{1+v^2-5 v^2}{5 v} \end{aligned}$

$\begin{aligned} & \frac{1}{8} \int \frac{8 \times 5 v d v}{1-4 v^2}=\int \frac{d x}{x} \\ & \frac{-5}{8} \ln \left|1-4 v^2\right|=\ln |x|+\ln c \\ & \Rightarrow \frac{5}{8} \ln \frac{\left|x^2-4 y^2\right|}{x^2}+\ln |x|=\ln c \\ & \left|\frac{x^2-4 y^2}{x^2}\right|^{5 / 8}|x|=c \\ & \frac{\left|x^2-4 y^2\right|^{5 / 8}}{|x|^{\frac{1}{4}}}=c \because y(1)=0 \Rightarrow c=1 \\ \end{aligned}$

$\begin{aligned} & \Rightarrow\left|x^2-4 y^2\right|^{5 / 8}=|x|^{\frac{1}{4}} \\ & \left|x^2-4 y^2\right|^5=x^2 \end{aligned}$

$\begin{aligned} & 2 y \frac{d y}{d x}+3=5 \frac{d y}{d x} \\ & 2 y d y+3 d x=5 d y \\ & y^2+3 x=5 y+\left.c\right|_{(0,1)} \\ & 1+0=5+c \\ & c=-4 \\ & y^2-5 y=-3 x-4 \\ & y^2-5 y+\frac{25}{4}-\frac{25}{4}=-3 x-4 \\ & \left(y-\frac{5}{2}\right)^2=-3 x+\frac{9}{4} \\ & \left(y-\frac{5}{2}\right)^2=-3\left(x-\frac{3}{4}\right) \\ & \left(\frac{3}{4}, \frac{5}{2}\right) \text { satisfies by } 2 x+3 y=9 \end{aligned}$

$\begin{aligned} & \sec y \frac{d y}{d x}+2 x \sin y=x^3 \cos y \\ & \Rightarrow \sec ^2 y \frac{d y}{d x}+2 x \tan y=x^3 \end{aligned}$

Let $z=\tan y$

$\begin{aligned} & \frac{d z}{d x}=\sec ^2 y \frac{d y}{d x} \\ & \Rightarrow \frac{d z}{d x}+2 x z=x^3 \end{aligned}$

$\begin{aligned} & \text { I.F. }=e^{x^2} \\ & \Rightarrow z . e^{x^2}=\int e^{x^2} \cdot x^3 d x+c \end{aligned}$

$\Rightarrow \tan y \cdot e^{x^2}=\frac{1}{2}\left(x^2 e^{x^2}-e^{x^2}\right)+c$

$\Rightarrow \tan (0) \cdot e=\frac{1}{2}(1 \cdot e-e)+c$

$\Rightarrow c=0$

$\Rightarrow \tan y=\frac{x^2-1}{2}$

$f(x)=\tan ^{-1}\left(\frac{x^2-1}{2}\right) \Rightarrow f(\sqrt{3})=\frac{\pi}{4}$

$\int_\limits0^a f(x) d x=e^{-a}+4 a^2+a-1$

Differentiate equation w.r.t. 'a'

$\begin{aligned} & f(a)=-e^{-a}+8 a+1 \\ & \Rightarrow f(x)=-e^{-x}+8 x+1 \end{aligned}$

And $y=c_1 f(x)+c_2$

$\begin{aligned} & y=c_1\left(-e^{-x}+8 x+1\right)+c_2 \\ & y^{\prime}=c_1\left(e^{-x}+8\right) \Rightarrow c_1=\frac{y^{\prime}}{e^{-x}+8} \end{aligned}$

$y^{\prime \prime}=-c_1 e^{-x}$

put value of $c_1$

$\begin{aligned} & \frac{d^2 y}{d x^2}=\frac{-\frac{d y}{d x} \cdot e^{-x}}{\left(e^{-x}+8\right)}=\frac{\frac{d y}{d x}}{\left(1+8 e^x\right)} \\ & \Rightarrow\left(1+8 e^x\right) \frac{d^2 y}{d x^2}+\frac{d y}{d x}=1 \end{aligned}$

$\begin{aligned} & \left(1+y^2\right) e^{\tan x} d x+\cos ^2 x\left(1+e^{2 \tan x}\right) d y=0 \\ & \frac{d y}{1+y^2}=-\frac{e^{\tan x} \cdot \sec ^2 x d x}{1+e^{2 \tan x}} \\ & \int \frac{d y}{1+y^2}=-\int \frac{e^{\tan x} \cdot \sec ^2 x d x}{1+e^{2 \tan x}} \end{aligned}$

Let $e^{\tan x}=t$

$\begin{aligned} & e^{\tan x} \cdot \sec ^2 x d x=d t \\ & \int \frac{d y}{1+y^2}=-\int \frac{d t}{1+t^2} \end{aligned}$

$\begin{aligned} & \tan ^{-1} y=-\tan ^{-1} t+c \\ & \tan ^{-1} y=-\tan ^{-1}\left(e^{\tan x}\right)+c \\ & \text { at } x=0, y=1 \\ & \tan ^{-1}(1)=-\tan ^{-1}(1)+c \\ & \frac{\pi}{4}=-\frac{\pi}{4}+c \\ & c=\frac{\pi}{2} \\ & \tan ^{-1} y=-\tan ^{-1}\left(e^{\tan x}\right)+\frac{\pi}{2} \end{aligned}$

Now, at $x=\frac{\pi}{4}$

$\begin{aligned} & \tan ^{-1} y=-\tan ^{-1}(e)+\frac{\pi}{2} \\ & \tan ^{-1} y=\cot ^{-1} e=\tan ^{-1} \frac{1}{e} \\ & \Rightarrow y=\frac{1}{e} \end{aligned}$

$\begin{aligned} & \frac{d y}{d x}=\frac{(2+\alpha) x-\beta y+2}{\beta x-2 \alpha y-(\beta \gamma-4 \alpha)} \\ & \beta x d y-2 \alpha y d y-(\beta \gamma-4 \alpha) d y \\ & =2 x d x+\alpha x d x-\beta y d x+2 d x \\ & \beta \int(x d y+y d y)-\alpha y^2-(\beta \gamma-4 x) y=x^2+\frac{\alpha x^2}{2}+2 x \\ & \beta x y-\alpha y^2-(\beta \gamma-4 \alpha) y=x^2+\frac{\alpha x^2}{2}+2 x \\ & \left(1+\frac{\alpha}{2}\right) x^2+\alpha y^2-\beta x y+2 x+(\beta \gamma-4 \alpha) y=0 \\ & \because \text { this represents circle passing through origin } \\ & \Rightarrow \beta=0 \text { and } 1+\frac{\alpha}{2}=\alpha \\ & \Rightarrow \alpha=2 \end{aligned}$

$\begin{aligned} & \therefore C: 2 x^2+2 y^2+2 x-8 y=0 \\ & x^2+y^2+x-4 y=0 \\ & \text { Radius }=\sqrt{\frac{1}{4}+4-0} \\ & \quad=\frac{\sqrt{17}}{2} \end{aligned}$

$\begin{aligned} & (2 x \log x) \frac{d y}{d x}+2 y=\frac{3}{x} \log x \\ & \frac{d y}{d x}+\frac{y}{x \log x}=\frac{3}{2 x^2} \\ & I F=e^{\int \frac{1}{x \log x}}=e^{\log (\log x)}=\log x \\ & y \times \log x=\int \log x \times \frac{3}{2 x^2} d x+C \\ & y \log x=\frac{3}{2}\left[\frac{-\log x}{x}-\frac{1}{x}\right]+C \\ & y\left(e^{-1}\right)=0 \\ & \Rightarrow 0=\frac{3}{2}[e-e]+C \\ & C=0 \\ & y \log x=\frac{-3}{2}\left[\frac{\log x+1}{x}\right] \\ & y(e) \rightarrow y(e) \times 1=\frac{-3}{2}\left[\frac{1+1}{e}\right]=-\frac{3}{e} \end{aligned}$

To determine $ y(0) $, we start by solving the differential equation given:

$ (1 + x^2) \frac{dy}{dx} + y = e^{\tan^{-1} x} $

First, we rewrite it in the standard form for a linear differential equation:

$ \frac{dy}{dx} + \frac{y}{1 + x^2} = \frac{e^{\tan^{-1} x}}{1 + x^2} $

Next, we find the integrating factor (I.F.):

$ \text{I.F.} = e^{\int \frac{1}{1 + x^2} dx} = e^{\tan^{-1} x} $

Multiply through by the integrating factor:

$ y \cdot e^{\tan^{-1} x} = \int e^{\tan^{-1} x} \cdot \frac{e^{\tan^{-1} x}}{1 + x^2} dx $

This simplifies to:

$ y \cdot e^{\tan^{-1} x} = \int \frac{e^{2 \tan^{-1} x}}{1 + x^2} dx $

Make the substitution $ \tan^{-1} x = t $, then $ \frac{1}{1 + x^2} dx = dt $:

$ \int e^{2t} dt = \frac{e^{2t}}{2} + \frac{C}{2} $

Rewrite in terms of $ x $:

$ y \cdot e^{\tan^{-1} x} = \frac{e^{2 \tan^{-1} x}}{2} + \frac{C}{2} $

Use the initial condition $ y(1) = 0 $:

$ 0 = \frac{e^{2 \cdot \tan^{-1} 1}}{2} + \frac{C}{2} $

Since $ \tan^{-1} 1 = \frac{\pi}{4} $:

$ 0 = \frac{e^{\pi/2}}{2} + \frac{C}{2} \implies C = -e^{\pi/2} $

Thus, the solution is:

$ y \cdot e^{\tan^{-1} x} = \frac{e^{2 \tan^{-1} x} - e^{\pi/2}}{2} $

Evaluating $ y(0) $:

$ y(0) = y \cdot 1 = \frac{1 - e^{\pi/2}}{2} $

Therefore,

$ y(0) = \frac{1}{2} (1 - e^{\pi/2}) $

Hence, the correct answer is:

Option B

$ \frac{1}{2}\left(1-e^{\pi / 2}\right) $

Equation of circle passing through origin & having centre at the line $y=x$ is

$\begin{aligned} & (x-t)^2+(y-t)^2=2 t^2 \\ & x^2+y^2+t^2+t^2-2 t x-2 t y=2 t^2 \\ & x^2+y^2=2 t(x+y) \end{aligned}$

Now differentiate

$\begin{aligned} & 2 x+2 y y^{\prime}=2 t\left(1+y^{\prime}\right) \\ & t=\frac{x+y y^{\prime}}{1+y^{\prime}} \end{aligned}$

Now, $x^2+y^2=2\left(\frac{x+y y^{\prime}}{1+y^{\prime}}\right)(x+y)$

$\begin{aligned} & x^2+y^2+x^2 \frac{d y}{d x}+y^2 \frac{d y}{d x} \\ & =2\left(x^2+x y+x y \frac{d y}{d x}+y^2 \frac{d y}{d x}\right) \\ & d x\left(x^2+y^2-2 x^2-2 x y\right)=d y\left(2 x y+2 y^2-x^2-y^2\right) \\ & d x\left(x^2-y^2+2 x y\right)=d y\left(x^2-y^2-2 x y\right) \end{aligned}$

$\begin{aligned} & \frac{d y}{d x}+2 y=\sin 2 x \\ & \text { IF }=e^{2 d x}=e^{2 x} \\ & y \cdot e^{2 x}=\int e^{2 x} \sin 2 x d x+c \\ & =\frac{e^{2 x}}{8}(2 \sin 2 x-2 \cos 2 x)+c \\ & y(0)=\frac{3}{4} \\ & \frac{3}{4}=\frac{1}{8}(-2)+c \Rightarrow c=1 \\ & \text { Put } x=\frac{\pi}{8} \\ & y=\frac{1}{8} \times 2\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}\right)+e^{-\pi / 4} \\ & y=e^{-\pi / 4} \end{aligned}$

$\frac{d y}{d x}+\frac{y\left(2 x^3+8 x\right)}{\left(x^2+4\right)^2}=\frac{2}{\left(x^2+4\right)^2}$

$\mathrm{IF}=e^{\int \frac{2 x^3+8 x}{\left(x^2+4\right)^2} d x}$

$\text { Let }\left(x^2+4\right)^2=t \quad \Rightarrow 2\left(x^2+4\right)(2 x) d x=d t$

$\begin{gathered} =e^{\int \frac{d t}{2 t}}=e^{\log \sqrt{t}}=\sqrt{t}=\left(x^2+4\right) \\ \therefore \quad y\left(x^2+4\right)=\int \frac{2}{x^2+4}+c \\ \Rightarrow y\left(x^2+4\right)=\tan ^{-1}\left(\frac{x}{2}\right)+c \\ y(0)=0 \end{gathered}$

$\begin{aligned} & \Rightarrow 0=0+c \quad \Rightarrow c=0 \\ & \text { put } x=2 \\ & y(8)=\frac{\pi}{4} \quad \Rightarrow \quad y=\frac{\pi}{32} \\ \end{aligned}$

$\begin{aligned} & \left(x^4+2 x^3+3 x^2+2 x+2\right) d y-\left(2 x^2+2 x+3\right) d x=0 \\ & \int d y=\int\left(\frac{2 x^2+2 x+3}{x^4+2 x^3+3 x^2+2 x+2} d x\right. \\ & \int d y=\int \frac{1}{x^2+1} d x+\int \frac{}{x^2+2 x+2} d x \\ & y=\tan ^{-1}(x)+\tan ^{-1}(1+x)+C \\ & y(-1)=\tan ^{-1}(-1)+\tan ^{-1}(1-1)+C \\ & y(-1)=-\frac{\pi}{4}+C=\left(\frac{-\pi}{4}\right)-\{\text { given }\} \\ & \Rightarrow C=0 \\ & \text { So, } y(x)=\tan ^{-1}(x)+\tan ^{-1}(1+x) \\ & y(0)=\tan ^{-1}(0)+\tan ^{-1}(1+0) \\ & y(0)=\frac{\pi}{4} \end{aligned}$

$\begin{aligned} & \frac{\mathrm{dT}}{\mathrm{dt}}=-\mathrm{k}(\mathrm{T}-80) \\ & \int_\limits{160}^{\mathrm{T}} \frac{\mathrm{dT}}{(\mathrm{T}-80)}=\int_\limits0^{\mathrm{t}}-\mathrm{Kdt} \\ & {[\ln |\mathrm{T}-80|]_{160}^{\mathrm{T}}=-\mathrm{kt}} \\ & \ln |\mathrm{T}-80|-\ln 80=-\mathrm{kt} \\ & \ln \left|\frac{\mathrm{T}-80}{80}\right|=-\mathrm{kt} \\ & \mathrm{T}=80+80 \mathrm{e}^{-\mathrm{kt}} \\ & 120=80+80 \mathrm{e}^{-\mathrm{k} .15} \\ & \frac{40}{80}=\mathrm{e}^{-\mathrm{k} 15}=\frac{1}{2} \\ & \therefore \mathrm{T}(45)=80+80 \mathrm{e}^{-\mathrm{k} .45} \\ & =80+80\left(\mathrm{e}^{-\mathrm{k} .15}\right)^3 \\ & =80+80 \times \frac{1}{8} \\ & =90 \end{aligned}$

$\begin{aligned} & \frac{d y}{d x}=\frac{\sin x+y \cos x}{\sin x \cdot \cos x\left(\frac{1}{\cos x}-\sin x \cdot \frac{\sin x}{\cos x}\right)} \\ & =\frac{\sin x+y \cos x}{\sin x\left(1-\sin ^2 x\right)} \\ & \frac{d y}{d x}=\sec ^2 x+y \cdot 2(\operatorname{cosec} 2 x) \\ & \frac{d y}{d x}-2 \operatorname{cosec}(2 x) \cdot y=\sec ^2 x \\ & \frac{d y}{d x}+p \cdot y=Q \end{aligned}$

$\text { I.F. }=\mathrm{e}^{\int p \mathrm{dx}}=\mathrm{e}^{\int-2 \operatorname{cosec}(2 \mathrm{x}) \mathrm{dx}}$

Let $2 \mathrm{x}=\mathrm{t}$

$2 \frac{\mathrm{dx}}{\mathrm{dt}}=1$

$\mathrm{dx}=\frac{\mathrm{dt}}{2}$

$=\mathrm{e}^{-\int \operatorname{cosec}(\mathrm{t}) \mathrm{dt}}$

$=\mathrm{e}^{-\ln \left|\tan \frac{t}{2}\right|}$

$=\mathrm{e}^{-\ln |\tan x|}=\frac{1}{|\tan x|}$

$\begin{aligned} & y(\text { IF })=\int Q(I F) d x+c \\ & \Rightarrow y \frac{1}{|\tan x|}=\int \sec ^2 x \cdot \frac{1}{|\tan x|}+c \\ & y \cdot \frac{1}{|\tan x|}=\int \frac{d t}{|t|}+c \quad \text { for } \tan x=t \\ & y \cdot \frac{1}{|\tan x|}=\ln |t|+c \\ & y=|\tan x|(\ln |\tan x|+c) \\ & \text { Put } x=\frac{\pi}{4}, y=2 \\ & 2=\ln 1+c \Rightarrow c=2 \\ & y=|\tan x|(\ln |\tan x|+2) \\ & y\left(\frac{\pi}{3}\right)=\sqrt{3}(\ln \sqrt{3}+2) \end{aligned}$

$\frac{\mathrm{dx}}{\mathrm{dy}}=\frac{\mathrm{x}}{\mathrm{y}}\left(\ln \left(\frac{\mathrm{x}}{\mathrm{y}}\right)+1\right)$

Let $\frac{x}{y}=t \Rightarrow x=t y$

$\begin{aligned} & \frac{d x}{d y}=t+y \frac{d t}{d y} \\ & t+y \frac{d t}{d y}=t(\ln (t)+1) \end{aligned}$

$\mathrm{y} \frac{\mathrm{dt}}{\mathrm{dy}}=\mathrm{t} \ln (\mathrm{t}) \Rightarrow \frac{\mathrm{dt}}{\mathrm{t} \ln (\mathrm{t})}=\frac{\mathrm{dy}}{\mathrm{y}}$

$\Rightarrow \int \frac{\mathrm{dt}}{\mathrm{t} \cdot \ln (\mathrm{t})}=\int \frac{\mathrm{dy}}{\mathrm{y}}$

$\Rightarrow \int \frac{d p}{p}=\int \frac{d y}{y} \quad$ let $\ln t=p$

$\frac{1}{\mathrm{t}} \mathrm{dt}=\mathrm{dp}$

$\begin{aligned} & \Rightarrow \ln p=\ln y+c \\ & \ln (\ln t)=\ln y+c \\ & \ln \left(\ln \left(\frac{x}{y}\right)\right)=\ln y+c \\ & \text { at } x=e, y=1 \\ & \ln \left(\ln \left(\frac{e}{1}\right)\right)=\ln (1)+c \Rightarrow c=0 \end{aligned}$

$\begin{aligned} & \ln \left|\ln \left(\frac{x}{y}\right)\right|=\ln y \\ & \left|\ln \left(\frac{x}{y}\right)\right|=e^{\ln y} \\ & \left|\ln \left(\frac{x}{y}\right)\right|=y \end{aligned}$

$\frac{d y}{d x}=2(x-1) \sin x+\left(x^2-2 x\right) \cos x$

Now both side integrate

$\begin{aligned} & y(x)=\int 2(x-1) \sin x d x+\left[\left(x^2-2 x\right)(\sin x)-\int(2 x-2) \sin x d x\right] \\ & y(x)=\left(x^2-2 x\right) \sin x+\lambda \\ & y(0)=0+\lambda \Rightarrow 2=\lambda \\ & y(x)=\left(x^2-2 x\right) \sin x+2 \\ & y(2)=2 \end{aligned}$

Differential equation :-

$\begin{aligned} & x \cos \frac{y}{x} \frac{d y}{d x}=y \cos \frac{y}{x}+x \\ & \cos \frac{y}{x}\left[x \frac{d y}{d x}-y\right]=x \end{aligned}$

Divide both sides by $\mathrm{x}^2$

$\cos \frac{y}{x}\left(\frac{x \frac{d y}{d x}-y}{x^2}\right)=\frac{1}{x}$

Let $\frac{y}{x}=t$

$\begin{aligned} & \cos \mathrm{t}\left(\frac{\mathrm{dt}}{\mathrm{dx}}\right)=\frac{1}{\mathrm{x}} \\ & \cos \mathrm{t~dt}=\frac{1}{\mathrm{x}} \mathrm{dx} \end{aligned}$

Integrating both sides

$\begin{aligned} & \sin \mathrm{t}=\ln |\mathrm{x}|+\mathrm{c} \\ & \sin \frac{\mathrm{y}}{\mathrm{x}}=\ln |\mathrm{x}|+\mathrm{c} \end{aligned}$

Using $\mathrm{y}(1)=\frac{\pi}{3}$, we get $\mathrm{c}=\frac{\sqrt{3}}{2}$

So, $\alpha=\sqrt{3} \Rightarrow \alpha^2=3$

$\begin{aligned} & \frac{d y}{d x}-\left(\frac{\sin 2 x}{1+\cos ^2 x}\right) y=\sin x \\ & \text { I.F. }=1+\cos ^2 x \\ & y \cdot\left(1+\cos ^2 x\right)=\int(\sin x) d x \\ & =-\cos x+C \\ & x=0, C=1 \\ & y\left(\frac{\pi}{2}\right)=1 \end{aligned}$

$\begin{aligned} & \left(x^2-4\right) d y-\left(y^2-3 y\right) d x=0 \\ & \Rightarrow \int \frac{d y}{y^2-3 y}=\int \frac{d x}{x^2-4} \\ & \Rightarrow \frac{1}{3} \int \frac{y-(y-3)}{y(y-3)} d y=\int \frac{d x}{x^2-4} \\ & \Rightarrow \frac{1}{3}(\ln |y-3|-\ln |y|)=\frac{1}{4} \ln \left|\frac{x-2}{x+2}\right|+C \\ & \Rightarrow \frac{1}{3} \ln \left|\frac{y-3}{y}\right|=\frac{1}{4} \ln \left|\frac{x-2}{x+2}\right|+C \end{aligned}$

$\begin{aligned} & \text { At } \mathrm{x}=4, \mathrm{y}=\frac{3}{2} \\ & \therefore \mathrm{C}=\frac{1}{4} \ln 3 \\ & \therefore \frac{1}{3} \ln \left|\frac{\mathrm{y}-3}{\mathrm{y}}\right|=\frac{1}{4} \ln \left|\frac{\mathrm{x}-2}{\mathrm{x}+2}\right|+\frac{1}{4} \ln (3) \\ & \text { At } \mathrm{x}=10 \\ & \frac{1}{3} \ln \left|\frac{\mathrm{y}-3}{\mathrm{y}}\right|=\frac{1}{4} \ln \left|\frac{2}{3}\right|+\frac{1}{4} \ln (3) \\ & \ln \left|\frac{\mathrm{y}-3}{\mathrm{y}}\right|=\ln 2^{3 / 4}, \forall \mathrm{x}>2, \frac{\mathrm{dy}}{\mathrm{dx}}<0 \\ & \text { as } \mathrm{y}(4)=\frac{3}{2} \Rightarrow \mathrm{y} \in(0,3) \\ & -\mathrm{y}+3=8^{1 / 4} \cdot \mathrm{y} \\ & \mathrm{y}=\frac{3}{1+8^{1 / 4}} \end{aligned}$

$\begin{aligned} & \frac{\mathrm{dx}}{\mathrm{dt}}+\mathrm{ax}=0 \\ & \frac{\mathrm{dx}}{\mathrm{x}}=-\mathrm{adt} \\ & \int \frac{\mathrm{dx}}{\mathrm{x}}=-\mathrm{a} \int \mathrm{dt} \\ & \ln |\mathrm{x}|=-\mathrm{at}+\mathrm{c} \\ & \mathrm{at} \mathrm{t}=0, \mathrm{x}=2 \\ & \ln 2=0+\mathrm{c} \\ & \ln \mathrm{x}=-\mathrm{at}+\ln 2 \\ & \frac{\mathrm{x}}{2}=\mathrm{e}^{-\mathrm{at}} \\ & \mathrm{x}=2 \mathrm{e}^{-\mathrm{at}} \quad \text{.... (i)} \end{aligned}$

$\begin{aligned} & \frac{d y}{d t}+b y=0 \\ & \frac{d y}{y}=-b d t \\ & \ln |y|=-b t+\lambda \\ & t=0, y=1 \\ & 0=0+\lambda \\ & y=e^{-b t} \quad \text{..... (ii)} \end{aligned}$

According to question

$\begin{aligned} & 3 \mathrm{y}(1)=2 \mathrm{x}(1) \\ & 3 \mathrm{e}^{-\mathrm{b}}=2\left(2 \mathrm{e}^{-\mathrm{a}}\right) \\ & \mathrm{e}^{\mathrm{a}-\mathrm{b}}=\frac{4}{3} \end{aligned}$

$\begin{aligned} & \text { For } x(t)=y(t) \\ & \begin{array}{r} 2 \mathrm{e}^{-a t}=e^{-b t} \\ 2=e^{(a-b) t} \\ 2=\left(\frac{4}{3}\right)^t \\ \log _{\frac{4}{3}} 2=t \end{array} \end{aligned}$

The given differential equation is

$\frac{d y}{d x} = y + 7$

This is a first order linear differential equation and can be solved using an integrating factor.

Rearrange the equation to the standard form of a linear differential equation :

$\frac{d y}{d x} - y = 7$

The integrating factor is $e^{-\int dx} = e^{-x}$.

Multiplying each side of the equation by the integrating factor gives :

$e^{-x} \frac{d y}{d x} - e^{-x} y = 7e^{-x}$

The left-hand side of the equation is the derivative of $(e^{-x}y)$ with respect to $x$. So we can write the equation as :

$\frac{d}{d x}(e^{-x}y) = 7e^{-x}$

Integrate both sides with respect to $x$ :

$e^{-x}y = -7e^{-x} + C$

Multiply both sides by $e^{x}$ to isolate $y$ :

$y = -7 + Ce^{x}$

So, the general solution to the differential equation is $y = -7 + Ce^{x}$.

Now, let's apply the initial conditions to find the particular solutions :

For $y_{1}(0)=0$, we substitute into the general solution and solve for $C$ :

$0 = -7 + C$

So, $C = 7$, and the solution for $y_{1}$ is $y_{1}(x) = 7e^{x} - 7$.

For $y_{2}(0)=1$, again substitute into the general solution:

$1 = -7 + C$

So, $C = 8$, and the solution for $y_{2}$ is $y_{2}(x) = 8e^{x} - 7$.

The two curves intersect when $y_{1}(x) = y_{2}(x)$. Setting these equal and solving for $x$ gives :

$7e^{x} - 7 = 8e^{x} - 7$

$e^{x} = 0$

But $e^{x} = 0$ has no solution, because the exponential function never equals zero.

So, the curves $y=y_{1}(x)$ and $y=y_{2}(x)$ do not intersect at any point.