Question 1: Palette 1/69

AP-EAPCET 2025 MCQ

If $y=A t^2+\frac{B}{t}$ ( $A, B$ are parameters) is general solution of the differential equation $f(t) y^{\prime \prime}(t)+g(t) y^{\prime}(t)+h(t) y=0$ then $2 f(t)+t^2 h(t)=$

A

$g(t)-h(t)$

B

$g(t)+f(t)$

C

$g(t) f(t)$

D

$(f(t))^{g( t)}$

Question 2: Palette 2/69

AP-EAPCET 2025 MCQ

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

A

$\log (2 x-y)=2 x+C$

B

$(2 x-y)^3+4 y=C$

C

$(2 x-y)^3+6 x=C$

D

$\log (2 x-y)=2 y+C$

Question 3: Palette 3/69

AP-EAPCET 2025 MCQ

The general solutions of the differential equation $x \log x d y=(x \log x-y) d x$ is

A

$(x-y) \log x+x=C$

B

$x-y=\frac{x}{\log x}+C$

C

$y-x=\frac{x}{\log x}+C$

D

$(y-x) \log x+x=C$

Question 4: Palette 4/69

AP-EAPCET 2025 MCQ

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

A

$\cos \left(\frac{y}{x}\right)=\log |x|+C$

B

$\cos \left(\frac{y}{x}\right)=\frac{1}{x}+C$

C

$\cos \left(\frac{x}{y}\right)=\log |y|+C$

D

$\cos \frac{y}{x}=\frac{2}{x}+C$

Question 5: Palette 5/69

AP-EAPCET 2025 MCQ

The general solution of the differential equation $\cos (x+y) d y=d x$ is

A

$y=\tan \left(\frac{x+y}{2}\right)+C$

B

$y=\sec \left(\frac{x+y}{2}\right)+C$

C

$y=x \sec \left(\frac{y}{x}\right)+C$

D

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

Question 6: Palette 6/69

AP-EAPCET 2025 MCQ

If $A x^3+B x y=4$ ( $A$ and $B$ are arbitrary constants) is the general solution of the differential equation $F(x) \frac{d^2 y}{d x^2}+G(x) \frac{d y}{d x}-2 y=0$, then $F(l)+G(l)=$

A

1

B

0

C

4

D

9

Question 7: Palette 7/69

AP-EAPCET 2025 MCQ

If $a$ and $b$ are arbitrary constants, then the differential equation corresponding to the family of curves $y=\tan (a x+b)$ is

A

$\left(1+x^2\right) y_2-2 y y_1+y=0$

B

$\left(1+y^2\right) y_2-2 y y_1^2=0$

C

$\left(1+x^2\right) y_2+2 y y_1^2=0$

D

$\left(1+y^2\right) y_2-2 y y_1^2+y=0$

Question 8: Palette 8/69

AP-EAPCET 2025 MCQ

The general solution of the differential equation $x y(y+2) d y+\left(y^3-1\right) d x=0$ is

A

$\log |x+2 y|+\frac{2}{\sqrt{3}} \tan ^{-1}\left(\frac{y-x}{\sqrt{3} x}\right)=C$

B

$\log |2 x-y|+\frac{2}{3} \tan ^{-1}\left(\frac{x-y}{\sqrt{3} x}\right)=C$

C

$\log |x y-x|+\frac{2}{\sqrt{3}} \tan ^{-1}\left(\frac{2 y+1}{\sqrt{3}}\right)=C$

D

$\log |x+y|+\frac{2}{3} \tan ^{-1}\left(\frac{x-2 y}{\sqrt{3 x}}\right)=C$

Question 9: Palette 9/69

AP-EAPCET 2025 MCQ

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

A

$(\sin 2 x) y=\sin ^2 x+C$

B

$\left(1+\sin ^2 x\right) y=\sin x-\frac{\sin ^3 x}{3}+C$

C

$\left(1+\sin ^2 x\right) y=\sin x+\frac{\sin ^3 x}{3}+C$

D

$(\sin 2 x) y=\sin x+\sin ^2 x+C$

Question 10: Palette 10/69

AP-EAPCET 2025 MCQ

If the slope of the tangent drawn at any point $(x, y)$ on a curve is $(x+y)$, then the equation of that curve is

A

$y=c e^x+1+x$

B

$y=c e^x-x$

C

$y=c e^{-x}-1-x$

D

$y=c e^x-1-x$

Question 11: Palette 11/69

AP-EAPCET 2025 MCQ

The solution of the differential equation $x^2(y+1) \frac{d y}{d x}+y^2(x+1)^2=0$, when $y(1)=2$, is

A

$\log \left|x^2 y\right|=\frac{2}{x}+\frac{1}{y}+x-1$

B

$\log \left|\frac{1}{4} x^2 y\right|=\frac{1}{x}+\frac{2}{y}+x-1$

C

$\log \left|\frac{1}{2} x^2 y\right|=\frac{1}{x}+\frac{1}{y}-x-\frac{1}{2}$

D

$\log \left|\frac{1}{3} x^2 y\right|=\frac{1}{x}+\frac{1}{y}-x+\frac{1}{2}$

Question 12: Palette 12/69

AP-EAPCET 2025 MCQ

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

A

$x^2-x y-y^2+3 x+3 y+c=0$

B

$x^2-x y-y^2-3 x-3 y+c=0$

C

$x^2+x y-y^2-3 x-3 y+c=0$

D

$x^2+x y+y^2+3 x-3 y+c=0$

Question 13: Palette 13/69

AP-EAPCET 2025 MCQ

If $x \log x \frac{d y}{d x}+y=\log x^2$ and $y(e)=0$, then $y\left(e^2\right)=$

A

0

B

1

C

$\frac{1}{2}$

D

$\frac{3}{2}$

Question 14: Palette 14/69

AP-EAPCET 2025 MCQ

If the order and degree of the differential equation $x \frac{d^2 y}{d x^2}=\left(1+\left(\frac{d^2 y}{d x^2}\right)^2\right)^{-1 / 2}$ are $k$ and $l$ respectively, then $k, l$ are the roots of

A

$x^2-5 x+6=0$

B

$x^2-3 x+2=0$

C

$x^2-7 x+12=0$

D

$x^2-6 x+8=0$

Question 15: Palette 15/69

AP-EAPCET 2025 MCQ

The equation of the curve passing through the point $(0, \pi)$ and satisfying the differential equation $y d x=\left(x+y^3 \cos y\right) d y$ is

A

$x=y^2 \sin y+y \cos ^2 y$

B

$x=y^2 \sin y+2 y \cos ^2 \frac{y}{2}$

C

$x=y^2 \sin y+y \cos ^2 \frac{y}{2}$

D

$x=y^2 \sin y-y \cos ^2 y$

Question 16: Palette 16/69

AP-EAPCET 2025 MCQ

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

A

$y \log (x+y)=c x$

B

$\log (x+y)=c y$

C

$x \log (x+y)=c y$

D

$\log (x+y)=c x$

Question 17: Palette 17/69

AP-EAPCET 2025 MCQ

The general solution of the differential equation $\sec (x-y+1) d y=d x$ is

A

$x+\cot \left(\frac{x-y+1}{2}\right)=C$

B

$x+\cot (x-y+1)=C$

C

$x-\cot \left(\frac{x-y+1}{2}\right)=C$

D

$x-\cot (x-y+1)=C$

Question 18: Palette 18/69

AP-EAPCET 2025 MCQ

The differential equation for which $y^2=4 a(x+a)$ ( $a$ is the parameter) is the general solution is

A

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

B

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

C

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

D

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

Question 19: Palette 19/69

AP-EAPCET 2025 MCQ

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

A

$5(y-x)+2 \log \left(\frac{y-2}{x-2}\right)=C$

B

$2(y-x)-5 \log \left(\frac{y-2}{x-2}\right)=C$

C

$2(y-x)+5 \log \left(\frac{y-2}{x-2}\right)=C$

D

$5(y-x)-2 \log \left(\frac{y-2}{x-2}\right)=C$

Question 20: Palette 20/69

AP-EAPCET 2025 MCQ

The differential equation of the family of circles passing through the origin and having centre on $X$-axis is

A

$\left(y^2+x^2\right) d x-2 y d y=0$

B

$\left(y^2-x^2\right) d x-2 x y d y=0$

C

$\left(y^2-x^2\right) d x+2 y d y=0$

D

$\left(y^2+x^2\right) d x+2 y d y=0$

Question 21: Palette 21/69

AP-EAPCET 2025 MCQ

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

A

$y-x=c x^2$

B

$\tan ^{-1}\left(\frac{y}{x}\right)=\log \left(c x \sqrt{x^2+y^2}\right)$

C

$x+y=c x^2$

D

$\tan ^{-1}\left(\frac{y}{x}\right)=\log \left(c \sqrt{x^2+y^2}\right)$

Question 22: Palette 22/69

AP-EAPCET 2025 MCQ

The general solution of the differential equation $\frac{d y}{d x}+\frac{\sec x}{\cos x+\sin x} y=\frac{\cos x}{1+\tan x}$ is

A

$(\cos x+\sin x) y=\sin x+C$

B

$(\cos x+\sin x) y=\cos x+C$

C

$(1+\tan x) y=\cos x+C$

D

$\sec x(\cos x+\sin x) y=\sin x+C$

Question 23: Palette 23/69

AP-EAPCET 2025 MCQ

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

A

$\log \left|\frac{y^2-2 x^2}{x^2}\right|+\sqrt{2} \log \left|\frac{y-\sqrt{2} x}{y+\sqrt{2} x}\right| +2 \sqrt{2} \log |x|=C $

B

$\sqrt{2} \log \left|\frac{y^2-2 x^2}{x^2}\right|+\log \left|\frac{y-\sqrt{2} x}{y+\sqrt{2} x}\right| +2 \sqrt{2} \log |x|=C $

C

$\sqrt{2} \log \left|\frac{y^2+2 x^2}{x^2}\right|+\log \left|\frac{y+\sqrt{2} x}{y-\sqrt{2} x}\right| +2 \sqrt{2} \log |x|=C $

D

$\log \left|\frac{2 x^2-y^2}{x^2}\right|+\sqrt{2} \log \left|\frac{y+\sqrt{2} x}{y-\sqrt{2} x}\right| +\log |x|=C $

Question 24: Palette 24/69

AP-EAPCET 2025 MCQ

If the degree of the differential equation corresponding to the family of curves $y=a x+\frac{1}{a}$ (where $a \neq 0$ is an arbitary constant) is $r$ and it's order is $m$. Then, the solution of $\frac{d y}{d x}=\frac{y}{2 x}, y(\mathrm{l})=\sqrt{r+m}$ is

A

$y=3^x$

B

$y^2=3 x$

C

$x^2=3 y$

D

$y=3 \log x$

Question 25: Palette 25/69

AP-EAPCET 2025 MCQ

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

A

$(\sec x+\tan x) y=x+\cos x+c$

B

$(1+\cos x) y=(x+c) \cos x-\cos ^2 x$

C

$(1+\sin x) y=(x+c) \cos x-\cos ^2 x$

D

$(\sec x+\tan x) y=x-\sin x+c$

Question 26: Palette 26/69

AP-EAPCET 2025 MCQ

The general solution of the differential equation $\frac{d y}{d x}+x y=4 x-2 y+8$ is

A

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

B

$y=8+c e^{-\frac{x^2}{2}-2 x}$

C

$y=c e^{-(x+2)^2}+x$

D

$y+2 x=c e^{-\frac{x}{2}-2 x}$

Question 27: Palette 27/69

AP-EAPCET 2025 MCQ

The general solution of the differential equation $\left(x+2 y^3\right) \frac{d y}{d x}-y=0, y>0$ is

A

$y=x^3+c y$

B

$x=y^3+c y$

C

$y(1-x y)=c x$

D

$x(1-x y)=c y$

Question 28: Palette 28/69

AP-EAPCET 2025 MCQ

The general solution of the differential equation $\frac{d y}{d x}+\frac{x+y+1}{x-3 y+5}=0$ is

A

$3(y-1)^2-2(x+2)(y-1)-(x+2)^2=C$

B

$x^2-3 y^2-4 x y-2 x-10 y=C$

C

$3(y+1)^2+2(x-2)(y+1)-(x-2)^2=C$

D

$x^2+3 y^2+4 x y+2 x+10 y=C$

Question 29: Palette 29/69

AP-EAPCET 2025 MCQ

The differential equation corresponding to the family of parabolas whose axis is along $x=1$ is

A

$\frac{d^2 y}{d x^2}-(x-1) \frac{d y}{d x}=0$

B

$(x-1) \frac{d^2 y}{d x^2}-\frac{d y}{d x}=0$

C

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

D

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

Question 30: Palette 30/69

AP-EAPCET 2025 MCQ

The general solution of the equation $\frac{d y}{d x}+\frac{1}{x} y=\frac{1}{x} e^x$

A

$y=x e^x+c$

B

$y=x e^x+c e^{-x}$

C

$y=\frac{e^x+c}{x}$

D

$y=\frac{e^{-x}+c x}{x}$

Question 31: Palette 31/69

AP-EAPCET 2025 MCQ

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 $

A

$\log x+\tan \frac{y}{x}=C$

B

$\log x+\cos \frac{y}{x}=C$

C

$\log x-\sin \frac{y}{x}=C$

D

$\log x-\cos \frac{y}{x}=C$

Question 32: Palette 32/69

AP-EAPCET 2024 MCQ

Among the options given below from which option a differential equation of order two can be formed ?

A

All circles passing through origin

B

All parabolas passing through origin and having focus on X-axis

C

All the lines passing through the origin

D

All the hyperbolas of the form $x^2-y^2=K^2$

Question 33: Palette 33/69

AP-EAPCET 2024 MCQ

The differential equation for which $a x+b y=1$ is general solution is

A

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

B

$y \frac{d^2 y}{d x^2}+x=1$

C

$\frac{d^2 y}{d x^2}=0$.

D

$\frac{d^3 y}{d x^3}=0$

Question 34: Palette 34/69

AP-EAPCET 2024 MCQ

The solution of the differential equation $e^x y d x+e^x d y+x d x=0$ is

A

$e^x+y x^2=c$

B

$2 y e^x+x^2=c$

C

$y e^x+x^2 e^y=c$

D

$e^x+x e^y=c$

Question 35: Palette 35/69

AP-EAPCET 2024 MCQ

The differential equation of the family of hyperbols having their centres at origin and their axes along coordinates axes is

A

$x y y_2+x y_1^2-y y_1=0$

B

$x y_2-x y y_1^2+y y_1=0$

C

$x y y_2+x y_1^2+y y_1=0$

D

$x y_2+x y_1^2-y_1=0$

Question 36: Palette 36/69

AP-EAPCET 2024 MCQ

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

A

$c x y^2=e^{\frac{x}{y}}$

B

$c x y^2 e^{\frac{x}{y}}=1$

C

$c x y e^{\frac{x}{y}}=1$

D

$c x y=e^{\frac{x}{y}}$

Question 37: Palette 37/69

AP-EAPCET 2024 MCQ

The general solution of the differential equation $(1+\tan y)(d x-d y)+2 x d y=0$ is

A

$e^x(y \cos x+\sin x)+\sin x=c$

B

$e^x(y \cos x+y \sin x-\sin x)+\cos x=0$

C

$e^y(x \cos y+x \sin y-\sin y)=c$

D

$e^y(x \cos y+x \sin y+\sin y)=c$

Question 38: Palette 38/69

AP-EAPCET 2024 MCQ

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

A

$y+\sqrt{x^2+y^2}=c x^2$

B

$y+\sqrt{x^2+y^2}=c x$

C

$x+\sqrt{x^2+y^2}=c y$

D

$x-\sqrt{x^2+y^2}=c y^2$

Question 39: Palette 39/69

AP-EAPCET 2024 MCQ

The sum of the order and degree of differential equation $x\left(\frac{d^2 y}{d x^2}\right)^{1 / 2}=\left(1+\frac{d y}{d x}\right)^{4 / 3}$

A

5

B

8

C

12

D

10

Question 40: Palette 40/69

AP-EAPCET 2024 MCQ

The differential equation formed by eliminating arbitrary constants $A, B$ from the equation $y=A \cos 3 x+B \sin 3 x$ is

A

$\frac{d^2 y}{d x^2}+y=0$

B

$\frac{d^2 y}{d x^2}+9 y=0$

C

$\frac{d^2 y}{d x^2}-9 y=0$

D

$\frac{d^2 y}{d x^2}-y=0$

Question 41: Palette 41/69

AP-EAPCET 2024 MCQ

If $\cos x \frac{d y}{d x}-y \sin x=6 x,\left(0 < x < \frac{\pi}{2}\right)$ and $y\left(\frac{\pi}{3}\right)=0$, then $y\left(\frac{\pi}{6}\right)=$

A

$\frac{-\pi^2}{4 \sqrt{3}}$

B

$\frac{-\pi^2}{2}$

C

$\frac{-\pi^2}{2 \sqrt{3}}$

D

$\frac{\pi^2}{2 \sqrt{3}}$

Question 42: Palette 42/69

AP-EAPCET 2024 MCQ

$\frac{d y}{d x}=\frac{y+x \tan \frac{y}{x}}{x} \Rightarrow \sin \frac{y}{x}=$

A

$c x^2$

B

$c x$

C

$c x^3$

D

$c x^4$

Question 43: Palette 43/69

AP-EAPCET 2024 MCQ

The differential equation formed by eliminating $a$ and $b$ from the equation $y=a e^{2 x}+b x e^{2 x}$ is

A

$y^{\prime \prime}-4 y^{\prime}-4 y=0$

B

$y^{\prime \prime}+4 y^{\prime}-4 y=0$

C

$y^{\prime \prime}+4 y^{\prime}+4 y=0$

D

$y^{\prime \prime}-4 y^{\prime}+4 y=0$

Question 44: Palette 44/69

AP-EAPCET 2024 MCQ

If $y=a^3 e^{y^2 x+c}$ is the general solution of a differential equation, where $a$ and $c$ are arbitrary constants and $b$ is fixed constant, then the order of differential equation is

A

1

B

2

C

3

D

4

Question 45: Palette 45/69

AP-EAPCET 2024 MCQ

The solution of differential equation $\left(x+2 y^3\right) \frac{d y}{d x}=y$ ls

A

$x=y(2 x y+c)$

B

$x=y\left(y^2+c\right)$

C

$y=x\left(x^2+0\right)$

D

$x y=\frac{y^4}{2}+c$

Question 46: Palette 46/69

AP-EAPCET 2024 MCQ

Order and degree of the differential equation $\frac{d^3 y}{d x^3}=\left[1+\left(\frac{d y}{d x}\right)^2\right]^{\frac{5}{2}}$, respectively are

A

5,2

B

3,5

C

3,2

D

2,3

Question 47: Palette 47/69

AP-EAPCET 2024 MCQ

Integrating factor of the differential equation $\sin x \frac{d y}{d x}-y \cos x=1$ is

A

$\sin x$

B

$\cos x$

C

$\sec x$

D

$\operatorname{cosec} x$

Question 48: Palette 48/69

AP-EAPCET 2024 MCQ

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

A

$\cos \frac{x}{y}=\log _6 x+c$

B

$\cos \frac{x}{y}=\log _e y+c$

C

$\cos \frac{y}{x}=\log _e x+c$

D

$\cos \frac{y}{x}=\log _e y+c$

Question 49: Palette 49/69

AP-EAPCET 2024 MCQ

The sum of the order and degree of the differential equation $\frac{d^4 y}{d x^4}=\left\{c+\left(\frac{d y}{d x}\right)^2\right\}^{3 / 2}$ is

A

4

B

6

C

5

D

8

Question 50: Palette 50/69

AP-EAPCET 2024 MCQ

$ \begin{aligned} &\text { The general solution of the differential equation }\\ &(x+y) y d x+(y-x) x d y=0 \text { is } \end{aligned} $

A

$x+y \log (c y)=0$

B

$\frac{y}{x}=\log (x y)+c$

C

$x+y \log (c x y)=0$

D

$\frac{y}{x}=\log (c x y)$

Question 51: Palette 51/69

AP-EAPCET 2024 MCQ

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

A

$x+2+(y+1) \log (y+1)^2=y+c$

B

$x+2+\log (y+1)^2=\frac{y}{y+1}+c$

C

$\frac{x}{y+1}=\log (y+1)^2+y+c$

D

$\frac{x+2}{y+1}+\log (y+1)^2=y+c$

Question 52: Palette 52/69

AP-EAPCET 2024 MCQ

The difference of the order and degree of the differential equation $\left(\frac{d^2 y}{d x^2}\right)^{-\frac{7}{2}}\left(\frac{d^3 y}{d x^3}\right)^2-\left(\frac{d^2 y}{d x^2}\right)^{-\frac{5}{2}}\left(\frac{d^4 y}{d x^4}\right)=0$ is

A

5

B

3

C

4

D

2

Question 53: Palette 53/69

AP-EAPCET 2024 MCQ

If $x d y+\left(y+y^2 x\right) d x=0$ and $y=1$ at $x=1$, then

A

$y=\frac{x}{1+\log x}$

B

$y=\frac{1+\log x}{x}$

C

$y=x(1+\log x)$

D

$y=\frac{1}{x(1+\log x)}$

Question 54: Palette 54/69

AP-EAPCET 2024 MCQ

The solution of $x d y-y d x=\sqrt{x^2+y^2} d x$ when $y(\sqrt{3})=1$ is

A

$y^2+\sqrt{x^2+y^2}=x^2$

B

$5 y-\sqrt{x^2+y^2}=x^2$

C

$y+\sqrt{x^2+y^2}=x^2$

D

$5 y^2-\sqrt{x^2+y^2}=x$

Question 55: Palette 55/69

AP-EAPCET 2024 MCQ

The differential equation representing the family of circles having their centres of Y -axis is $\left(y_1=\frac{d y}{d x}\right.$ and $\left.y_2=\frac{d^2 y}{d x^2}\right)$

A

$y_2=y\left(y_1^2+1\right)$

B

$y_2=x y\left(y_1^2+1\right)$

C

$x_2=y_1\left(y_1^2+1\right)$

D

$x y_2=y\left(y_1^2+1\right)$

Question 56: Palette 56/69

AP-EAPCET 2024 MCQ

The general solution of the differential equation $\left(\sin y \cos ^2 y-x \sec ^2 y\right) d y=(\tan y) d r$, is

A

$\tan y=3 x \cos ^3 y+c$

B

$x(\sec y+\tan y)=\cos ^2 y+c$

C

$y \sin y=x^2 \cos ^2 y+c$

D

$3 x \tan y+\cos ^3 y=c$

Question 57: Palette 57/69

AP-EAPCET 2024 MCQ

The general solution of the differential equation $(x-y-1) d y=(x+y+1) d x$ is

A

$\tan ^{-1}\left(\frac{y+1}{x}\right)-\frac{1}{2} \log \left(x^2+y^2+2 y+1\right)=0$

B

$(x-y)+\log (x+y)=c$

C

$y^2-x^2+x y-3 y-x=c$

D

$(x-y-1)^2(x+y+1)^3=c$

Question 58: Palette 58/69

AP-EAPCET 2022 MCQ

The general solution of the differential equation $\frac{d y}{d x}=\cos ^2(3 x+y)$ is $\tan ^{-1}\left(\frac{\sqrt{3}}{2} \tan (3 x+y)\right)=f(x)$. Then, $f(x)=$

A

$2 \sqrt{3}(x+C)$

B

$x+C$

C

$\frac{x+C}{2 \sqrt{3}}$

D

$\frac{\sqrt{3}}{2}(x+C)$

Question 59: Palette 59/69

AP-EAPCET 2022 MCQ

If the general solution of the differential equation $\cos ^2 x \frac{d y}{d x}+y=\tan x$ is $y=\tan x-1+C e^{-\tan x}$ satisfies $y\left(\frac{\pi}{4}\right)=1$, then $C=$

A

e

B

1

C

$-$1

D

$\frac{1}{e}$

Question 60: Palette 60/69

AP-EAPCET 2022 MCQ

Assertion (A) Order of the differential equations of a family of circles with constant radius is two.

Reason (R) An algebraic equation having two arbitrary constants is general solution of a second order differential equation.

A

A and R are true, R is the correct explanation to A

B

A is true, R is false

C

A and R are true, R is not the correct explanation to A

D

A is false, R is true

Question 61: Palette 61/69

AP-EAPCET 2022 MCQ

If $l$ and $m$ are order and degree of a differential equation of all the straight lines at constant distance of $P$ units from the origin, then $l m^2+l^2 m=$

A

2

B

6

C

12

D

30

Question 62: Palette 62/69

AP-EAPCET 2022 MCQ

If $2 x-y+C \log (|x-2 y-4|)=k$ is the general solution of $\frac{d y}{d x}=\frac{2 x-4 y-5}{x-2 y+2}$, then $C=$

A

4

B

2

C

3

D

$-$4

Question 63: Palette 63/69

AP-EAPCET 2022 MCQ

By eliminating the arbitrary constants from $y=(a+b) \sin (x+c)-d e^{x+e+f}$, then differential equation has order of

A

6

B

4

C

3

D

5

Question 64: Palette 64/69

AP-EAPCET 2022 MCQ

If the solution of $\frac{d y}{d x}-y \log _e 0.5=0, y(0)=1$, and $y(x) \rightarrow k$, as $x \rightarrow \infty$, then $k=$

A

$\infty$

B

$-1$

C

1

D

0

Question 65: Palette 65/69

AP-EAPCET 2022 MCQ

$y=A e^x+B e^{-2 x}$ satisfies which of the following differential equations?

A

$\frac{d^2 y}{d x^2}-\frac{d y}{d x}+2 y=0$

B

$\frac{d^2 y}{d x^2}-2 \frac{d y}{d x}-y=0$

C

$\frac{d^2 y}{d x^2}-2 \frac{d y}{d x}+y=0$

D

$\frac{d^2 y}{d x^2}+\frac{d y}{d x}-2 y=0$

Question 66: Palette 66/69

AP-EAPCET 2021 MCQ

If $y=\sin (\sin x)$ and $y^{\prime \prime}+f(x) \cdot y^{\prime}+g(x) \cdot y=0$, then $f(x) \cdot g(x)$ is equal to

A

$\frac{1}{2} \sin (2 x)$

B

$\frac{1}{2} \cos (2 x)$

C

$\sin (2 x)$

D

$\cos (2 x)$

Question 67: Palette 67/69

AP-EAPCET 2021 MCQ

The equation of the curve passing through the point $\left(0, \frac{\pi}{4}\right)$ and satisfying the differential equation $\left(e^x \tan y\right) d x\left.+\left(1+e^x\right) \sec ^2 y\right) d y=0$ is given by

A

$\left(1+e^x\right) \tan y=2$

B

$1+e^x=2 \tan y$

C

$1+e^x=2 \sec y$

D

$\left(1+e^x\right) \tan y=k$

Question 68: Palette 68/69

AP-EAPCET 2021 MCQ

The solution of the differential equation $2x\left(\frac{dy}{dx}\right)-y=4$ represents a family of

A

ellipse

B

parabola

C

straight line

D

circle

Question 69: Palette 69/69

AP-EAPCET 2021 MCQ

The solution of the differential equation $\frac{d^2 y}{d x^2}+y=0$ is

A

$y=3 \sin x+4 \cos x$

B

$y=x^2$

C

$y=x+2$

D

$y=\log x$