Bernoulli's Equation

Q. Solution of the differential equation $2xy\frac{dy}{dx}=x^2+3y^2$ is :
(where $c$ is integration constant)

1. $x^3+y^2=c{x}^2$
2. $\frac{x^2}{2}+\frac{y^3}{2}=y^2+c$
3. $x^2+y^2=|c{x}^3|$
4. $|x^2-y^2|=c{x}^3$

Q.

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

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

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

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

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

Q. A function $y=f\left(x\right)$ satisfies $x{f}^{\prime }\left(x\right)-2f\left(x\right)=x^4{f}^2\left(x\right),\forall x>0$ and $f\left(1\right)=-6.$ Then the value of $f^{\prime }\left(3^{1/5}\right)$ is

Q.

If y(t) is a solution of $(1+t)\frac{dy}{dt}-ty=1$ and y(0)=-1, then show that $y\left(1\right)=-\frac{1}{2}.$

Q. The solution of the differential equation $\frac{dy}{dx}+x^{5}y=x^5{y}^7$ is
(where $c$ is integration constant)

1. $\ln |y^6-1|=x^6+c$
2. $\ln |y^{-6}-1|=x^6+c$
3. $\ln |y^{-6}+1|=x^{-6}+c$
4. $\ln |y^{-6}-1|=x^{-6}+c$

Q. A continuous function $f:R\to R$ satisfies the differential equation $f\left(x\right)=(1+x^2)(1+\underset{0}{\overset{x}{\int }}\frac{f^2\left(t\right)}{1+t^2}dt).$ If area of triangle formed by tangent drawn to the curve $y=f\left(x\right)$ at $x=1$ with the co-ordinates axis is $△,$ then the value of $\left[\frac{△}{3}\right]$ is
[Note: [K] denotes the greatest integer less than or equal to $K$.]

Q. Solution of the differential equation $\sin y\cdot \frac{dy}{dx}=\cos y(1-x\cos y)$ is:
(where $C$ is integration constant)

1. $\sec y=(1+x)e^x+C{e}^{-x}$
2. $\sec y=(1-x)+C{e}^x$
3. $\sec y=(1+x)+C{e}^x$
4. $\cos y=(1+x)+C{e}^x$

Q. The solution of differential equation coty dx=x dy is ______ .

Q. The general solution of the differential equation $\frac{dy}{dx}=\frac{x^2+xy+y^2}{x^2}$ is
(where $c$ is constant of integration)

1. ${\tan }^{-1}\left(\frac{x}{y}\right)=\ln |y|+c$
2. ${\tan }^{-1}\left(\frac{y}{x}\right)=\ln |x|+c$
3. ${\tan }^{-1}\left(\frac{x}{y}\right)=\ln |x|+c$
4. ${\tan }^{-1}\left(\frac{y}{x}\right)=\ln |y|+c$

Q. Prove that $y=\frac{4\sin \theta }{(2+\cos \theta )}-\theta$ is an increasing function of $\theta$ in $[0,\frac{\pi }{2}]$

Q. If $y\left(x\right)$ satisfies the differential equation $\frac{dy}{dx}=\sin 2x+3y\cot x$ and $y\left(\frac{\pi }{2}\right)=2,$ then which of the following statements is(are) CORRECT ?

1. $y\left(\frac{\pi }{6}\right)=0$
2. $y^{\prime }\left(\frac{\pi }{3}\right)=\frac{9-3\sqrt{2}}{2}$
3. $y\left(x\right)$ strictly increases in interval $(\frac{\pi }{6},\frac{\pi }{3})$
4. The value of definite integral $\underset{-\pi /2}{\overset{\pi /2}{\int }}y\left(x\right)dx$ equals $\pi$

Q. The general solution of the differential equation $\frac{dy}{dx}=ytanx-y^{2}secx$ is

1. tan x = (c + sec x)y
2. sec y = (c + tan y )x
3. sec x = (c + tan x)y
4. None of these

Q. The general solution of $\frac{dy}{dx}-\frac{y}{x}=\frac{y^2}{x^2}$ is

1. $\frac{x}{y}=-ln{x}^2+C$
2. $\frac{x}{y}=-ln\frac{x}{2}+C$
3. $\frac{x}{y}=-lnx+C$
4. $Noneofthese$

Q. The general solution of the differential equation $\frac{dy}{dx}=ytanx-y^{2}secx$ is

1. tan x = (c + sec x)y
2. None of these
3. sec y = (c + tan y )x
4. sec x = (c + tan x)y

Q. Let $y=y\left(x\right)$ be the solution of the differential equation $\frac{dy}{dx}=(y+1)\left(\right(y+1)e^{\frac{x^2}{2}}-x),0<x<2,$ such that $y\left(2\right)=0.$ Then the value of $\frac{dy}{dx}$ at $x=1$ is

1. $-\frac{e^{3/2}}{(e^2+1{)}^2}$
2. $-\frac{e^{1/2}}{(e^2+1)}$
3. $\frac{e^{1/2}}{(e^2+1{)}^2}(e^{1/2}-1)$
4. $\frac{e^{3/2}}{(e^2+1{)}^2}(e^{3/2}+1)$

Q. A homogeneous differential equation of the form $\frac{dx}{dy}=h\left(\frac{x}{y}\right)$ can be solved by making the substitution
(a) y = vx
(b) v = yx
(c) x = vy
(d) x = v

Q. If $\frac{dy}{dx}-y=y^2(\sin x+\cos x)$ with $y\left(0\right)=1$, then the value of $y\left(\pi \right)$ is

1. $-e^{\pi }$
2. $e^{-\pi }$
3. $-e^{-\pi }$
4. $e^{\pi }$

Q. $\mathrm{Which}\mathrm{of}\mathrm{the}\mathrm{following}\mathrm{is}\mathrm{the}\mathrm{solution}\mathrm{of}\mathrm{the}\mathrm{differential}\mathrm{equation}x\frac{\mathrm{dy}}{\mathrm{dx}}+y={\mathrm{xy}}^3?$

1. $y^2=\frac{1}{2x+{\mathrm{cx}}^2}$
2. $y^{}=\frac{1}{2x+{\mathrm{cx}}^2}$
3. $y^2=2x+{\mathrm{cx}}^2$
4. $y^3=\frac{1}{2x+{\mathrm{cx}}^2}$

Q. Which of the following transformations reduce the differential equation $\frac{dz}{dx}+\frac{z}{x}\log z=\frac{z}{x^2}{\left(\log z\right)}^2$ into the form $\frac{du}{dx}+P\left(x\right)u=Q\left(x\right)$
(a) u = log x
(b) u = e^z
(c) u = (log z)⁻¹
(d) u = (log z)²

Q. Solve the differential equation $(x^2-y^2)dx-xydy=0$.

1. $y^2(x^2-2y^2)=k$
2. $y^2(2x^2-y^2)=k$
3. $x^2(2x^2-y^2)=k$
4. $x^2(x^2-2y^2)=k$

Q. Solve the differential equation: $\frac{dy}{dx}=e^{x-y}(e^x-e^y).$

1. $e^y=(e^x+1)+c{e}^{-e^x}.$
2. None of these.
3. $e^y=(e^x-1)+c{e}^{-e^x}.$
4. $e^y=(e^x-1)-c{e}^{-e^x}.$

Q. The solution of the differential equation $\frac{dy}{dx}+1=e^{x+y}$, is
(a) (x + y) e^{x + y} = 0
(b) (x + C) e^{x + y} = 0
(c) (x − C) e^{x + y} = 1
(d) (x − C) e^{x + y} + 1 =0

Q.

For the given differential equation find the general solution.

$co{s}^{2}dx\frac{dy}{dx}+y=tanx(0\le x<\frac{pi}{2})$

Q. The point on the curve $3y=6x-5x^3$, the normal at which passes through the origin is

1. $\left(\frac{1}{3},1\right)$
2. none of these
3. $(2,\frac{-28}{3})$
4. $(1,\frac{1}{3})$

Q. Show that Ax² + By² = 1 is a solution of the differential equation x $\left\{y\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2\right\}=y\frac{dy}{dx}$.

Q. Solution of the differential equation $y+\frac{d}{dx}\left(xy\right)=x(\sin x+\ln |x|),$ is:
(where $c$ is integration constant)

1. $y=\cos x+\frac{2}{x}\sin x+\frac{2}{x^2}\sin x+\frac{x}{3}\ln |x|-\frac{x}{9}+\frac{c}{x^2}$
2. $y=-\sin x+\frac{2}{x}\cos x+\frac{2}{x^2}\cos x+\frac{x}{3}\ln |x|-\frac{x}{9}+\frac{c}{x^2}$
3. $y=-\cos x+\frac{2}{x}\sin x+\frac{2}{x^2}\cos x+\frac{x}{3}\ln |x|-\frac{x}{9}+\frac{c}{x^2}$
4. $y=-\cos x+\frac{3}{x}\sin x+\frac{4}{x^2}\cos x+\frac{x}{3}\ln |x|-\frac{x}{9}+\frac{c}{x^2}$

Q. The solution of the differential equation y₁ y₃ = y₂² is
(a) x = C₁ e^C₂y + C₃
(b) y = C₁ e^C₂x + C₃
(c) 2x = C₁ e^C₂y + C₃
(d) none of these

Q. The solution of the differential equation $\frac{dy}{dx}+\frac{1}{x}\tan y=\frac{\tan y\sin y}{x^2}$ is

1. $\frac{1}{y}cosecx=\frac{1}{2x^2}+k$
2. $\frac{1}{x}cosecy=\frac{1}{2x^2}+k$
3. $\frac{1}{x}\cos y=\frac{1}{2x^2}+k$
4. $\frac{1}{x}\cot y=\frac{1}{2x^2}+k$

Q. The solution of the differential equation $\frac{dy}{dx}+\frac{y}{2}\sec x=\frac{\tan x}{2y},$ where $0\le x\le \frac{\pi }{2},$ and $y\left(0\right)=1,$ is given by:

1. $y^2=1+\frac{x}{\sec x+\tan x}$
2. $y=1-\frac{x}{\sec x+\tan x}$
3. $y=1+\frac{x}{\sec x+\tan x}$
4. $y^2=1-\frac{x}{\sec x+\tan x}$

Q.

For the following question verify that the given function (explicit or implicit) is a solution of the corresponding differential equation.

$x+y=ta{n}^{-1}yand{y}^2{y}^{\prime }+y^2+1=0$