If the general solution of $\left(1+y^2\right) d x=\left(\tan ^{-1} y-x\right) d y$ is $x=f(y)+c e^{-\tan ^{-1} y}$, then $f(y)=$

If $y=f(x)$ is the solution of the differential equation $\left(1+\cos ^2 x\right) f^{\prime}(x)-4 \sin 2 x-f(x) \sin 2 x=0$ when $f(0)=0$, then $f\left(\frac{\pi}{3}\right)=$

The differential equation corresponding to the family of ellipses $\frac{x^2}{a^2}+\frac{y^2}{4}=1$, where ' $a$ ' is an arbitrary constant is

The general solution of the differential equation $\frac{d y}{d x}+(\sec x \operatorname{cosec} x) y=\cos ^2 x$

If the differential equation having $y=A e^x+B \sin x$ as its general solution is $f(x) \frac{d^2 y}{d x^2}+g(x) \frac{d y}{d x}+h(x) y=0$, then $f(x)+g(x)+h(x)=$

The differential equation of a family of hyperbolas whose axes are parallel to coordinate axes, centres lie on the line $y=2 x$ and eccentricity is $\sqrt{3}$ is

The general solution of the differential equation $\left(x^3-y^3\right) d x=\left(x^2 y-x y^2\right) d y$ is

The substitution required to reduce the differential equation $t^2 d x+\left(x^2-t x+t^2\right) d t=0$ to a differential equation which can be solved by variables separable method is

The equation which represents the system of parabolas whose axis is parallel to $Y$-axis satisfies the differential equation.

If $\cos x \frac{d y}{d x}=y \sin x-1, x \neq(2 n+1) \frac{\pi}{2}, n \in Z$ is the differential equation corresponding to the curve $y=f(x)$ and $f(0)=1$, then $f(x)$

The general solution of the differential equation $2 d x+d y=(6 x y+4 x-3 y) d x$ is

The order and degree of the differential equation

$ \frac{d y}{d x}=\left(\frac{d^{2} y}{d x^{2}}+2\right)^{\frac{1}{2}}+\frac{d^{2} y}{d x}+5 \text { are respectively } $

If $a$ and $b$ are the arbitrary constants, then the differential equation corresponding to the family of curves given by $y=x[a \cos (\log x)+b \sin (\log x)]$ is

If the solution for the differential equation $y^2 d x+\left(x^2-x y-y^2\right) d y=0$ at $(2,1)$ is $x+y=k\left(x y^2-y^3\right)$, then $k=$

The general solution of the differential equation $\frac{d y}{d x}+\frac{y}{x}=x^2$ is

If the order and degree of the differential equation corresponding to the family of curves $y^2=4 a(x+a)(a$ is parameter) are $m$ and $n$ respectively, then $m+n^2=$

If the solution of the differential equation $\frac{d y}{d x}=\frac{2 x+3 y}{3 x-2 y}$ is $y=x \tan (f(x))+C$, then $f(x)=$

The general solution of the differential equation $\left(x^2+2\right) d y+2 x y d x=e^x\left(x^2+2\right) d x$ is

The general solution of the differential equation $(3 x-4 y)(d x-3 d y)+(6 d x-4 d y)=0$ is

The general solution of the differential equation $(\sec x+\tan x) \frac{d y}{d x}+\left(\sec ^2 x+\sec x \tan x\right) y=1$ is

If $A$ and $B$ are arbitrary constants, then the differential equation having $y=A e^x+B \sin 2 x$ as its general solution is

The general solution of the differential equation $\frac{d y}{d x}=\sin (x-y)+\cos (x-y)$ is

The general solution of the differential equation $x^2 d y-\left(x y-y^2\right) d x=0$ is

The degree and order of the differential equation of the family of parabolas whose axis is the $X$-axis, are respectively

The general solution of the differential equation $\left(x \sin \frac{y}{x}\right) d y=\left(y \sin \frac{y}{x}-x\right) d x$ is

$f\left(x, y, c_1, c_2\right)=0$ is an equation containing two arbitrary constants $c_1$ and $c_2$. If the differential equation having $f\left(x, y, c_1, c_2\right)=0$ as its general solution is of $k$ th order, then the differential equation corresponding to $x^k+y^k=c^2$ ( $c$ is an arbitrary constant) is

If $l$ and $m$ are respectively the order and the degree of the differential equation $f(x) y^{\prime \prime}+g(x) y^{\prime}=\frac{4 y}{x}$ whose general solution is $y=a x^2+b x^2 \log x$, then $f(m)+g(m)=$

The general solution of the differential equation $d x=(2 x+3 y-4) d y$ is

The number of arbitrary constants that appear in the general solution of the differential equation $\left(\frac{d^4 y}{d x^4}+\frac{d^2 y}{d x^2}\right)^{3 / 2}=5 \frac{d^3 y}{d x^3}$ is

Assertion (A) The degree of the differential equation $y^{\prime \prime}+2 x y^{\prime}+\log _e\left(\frac{d y}{d x}\right)=0$ is 2 .

Reason (R) The degree of a differential equation is the highest degree of the highest order derivative occurring in the equation, after the equation is expressed in the form of a polynomial in differential coefficients. The correct option among the following

Let $S$ be the family of curves given by the general solution of the differential equation $\frac{y^2 e^{-1 / y}}{\sqrt{x}} d x-2 \sec \sqrt{x} d y=0$. Then, the equation of the curve belonging to $S$ and passing through $\left(\pi^2, 1\right)$ is

Statement I The differential equation corresponding to the family of circles having their centres on $Y$-axis and fixed radius $k$ is $\left(x^2-k^2\right)\left(\frac{d y}{d x}\right)^2+x^2=0$

Statement II The differential equation corresponding to the family of circles passing through the origin and having their centres on $X$-axis is $x^2-y^2+2 x y \frac{d y}{d x}=0$

Which of the above statements is (are) true?

If $m$ and $n$ are respectively the order and the degree of the differential equation representing the family of curves $y^2-5 a x-5 a^{3 / 2}=0(a>0$ is a parameter), then the value of $m-n$ is

The general solution of $\left(\left(1+x^2\right) y \sin x-2 x y\right) d x-\log y^{1+x^2} d y=0$ is

The equation of any member of the family of all the ellipses whose axes are along the coordinate axes satisfies the differential equation

The degree of the differential equation $\left(\frac{d^2 y}{d x^2}\right)^{\frac{4}{3}}+x\left(\frac{d y}{d x}\right)^2-y \cos \left(\frac{d y}{d x}\right)=0$ is

The general solution of the differential equation $\frac{d y}{d x}=\frac{2 x-3 y+5}{6 x-9 y+7}$ is

The differential equation corresponding to the family of curves given by $a x^2+b y^2=1$, where $a$ and $b$ are arbitrary constants is

For the differential equation

$ \sqrt{\frac{d^2 y}{d x^2}}=\sqrt[3]{\left[y \frac{d y}{d x}+x \sin \left(\frac{d y}{d x}\right)\right]^2} $

The general solution of the differential equation $\frac{d y}{d x}=\frac{x y+x-2 y-2}{x y-2 x+y-2}$ is

The differential equation of the family of circles with fixed radius $r$ units and centre on the line $y=3$, is

The degree of the differential equation

$ x\left(\frac{d^2 y}{d x^2}\right)^{1 / 3}+2 x^2\left(\frac{d^2 y}{d x^2}\right)^{5 / 3}+7 \frac{d y}{d x}+y=0 $

The curve that satisfies the differential equation $x y d y-\left(1+y^2\right) d x=0$ passes through $(1,0)$ and intersects the curve $x^2+3 y^2=3$ at an angle $\theta$. Then, $\frac{2 \theta}{\pi}=$

If $\alpha$ and $\beta$ are respectively the order and degree of the differential equation for which $a x^2+b y^2=1$ is the general solution, then the eccentricity of the ellipse $\alpha x^2+\beta y^2=1$ is

The solution of the differential equation $x d y-y d x=\sqrt{x^2+y^2} d x$, given that $y=1$ when $x=\sqrt{3}$, is

If the solution $y(x)$ of the differential equation $\sin x \frac{d y}{d x}+y \cos x=e^{2 x}, x \in(0, \pi)$ satisfies $y\left(\frac{\pi}{2}\right)=0$, then $y\left(\frac{\pi}{6}\right)=$

The order and degree of the differential equation $\frac{d^2 y}{d x^2}+y+\left(\frac{d y}{d x}-\frac{d^3 y}{d x^3}\right)^{3 / 2}=0$, are respectively.

The general solution of the differential equation $\frac{d y}{d x}=\frac{2 x-3 y+4}{3 x+2 y-7}$ is

The general solution of $\frac{d y}{d x}=\frac{x+y+1}{y-x+1}$ is

If $y=e^{a x}(\cos b x+\sin b x)$ satisfies the equation $\frac{d^2 y}{d x^2}-K \frac{d y}{d x}+L y=0$, then $L+b K=$

Let $f:[2,5] \rightarrow \mathbf{R}$ be a differentiatiable function and $\frac{f(5)}{f(2)}=1$. If there is a $c \in(2,5)$ such that $c f^{\prime}(c)=2 f(c)-2 c^3$, then $f(x)=$

Let $f:[2,5] \rightarrow \mathbf{R}$ be a differentiatiable function and $\frac{f(5)}{f(2)}=1$. If there is a $c \in(2,5)$ such that $c f^{\prime}(c)=2 f(c)-2 c^3$, then $f(x)=$

The differential equation for which $y=a x^2+b x+c$ is the general solution is

The general solution of the differential equation

$(3 y-7 x+7) d x+(7 y-3 x+3) d y=0$ is

The general solution of the differential equation $(3 y-7 x+7) d x+(7 y-3 x+3) d y=0$ is

The general solution of the differential equation $x \cos \frac{y}{x}(y d x+x d y)=y \sin \frac{y}{x}(x d y-y d x)$ is

If the family of curves $y=a e^{4 x}+b e^{-x}$, where $a, b$ are arbitrary constants represents the general solution of the differential equation

$ f\left(x, y \frac{d y}{d x}, \frac{d^2 y}{d x^2}\right)=0, \text { then } \frac{d f}{d x}= $

If the length of the sub tangent at any point $p(x, y)$ on a curve $f(x, y)=0$ is $x+7 y^2$, then $f(x, y)=$

If the general solution of the differential equation $(y-x+1) d y-(y+x+2) d x=0$ is $f(x, y, c)=0$, then the value of $c$ such that $f(1,1, c)=0$ is