Equations Reducible to Standard Forms

Q. The general solution of the differential equation $(y^2–x^3)dx–xydy=0(x\ne 0)$ is : (where $c$ is a constant of integration)

1. $y^2+2x^2+c{x}^3=0$
2. $y^2-2x^3+c{x}^2=0$
3. $y^2+2x^3+c{x}^2=0$
4. $y^2–2x^2+c{x}^3=0$

Q. The general solution of $y^{2}dx+(x^2-xy+y^2)dy=0$ is

1. $ta{n}^{-1}\left(\frac{x}{y}\right)+logy+c=0$
   
2. $2ta{n}^{-1}\left(\frac{x}{y}\right)+logx+c=0$
   
3. $log(y+\sqrt{x^2+y^2})+logy+c=0$
   
4. $sin{h}^{-1}\left(\frac{x}{y}\right)+logy+c=0$

Q.

Which of the following is a homogeneous differential equation?
(a) (4x +6y+5)dy-(3y+2x+4)dx=0
(b) $xydx-(x^3+y^3)dy=0$
(c) $(x^3+2y^2)dx+2xydy=0$
(d) $y^{2}dx+(x^2-xy-y^2)dy=0$

Q. If the tangent to the curve xy + ax + by = 0 at (1, 1) is inclined at an angle $ta{n}^{-1}2$ with x-axis, then

1. a = – 1, b = – 2
2. a = – 1, b = 2
3. a = 1, b = 2
4. a = 1, b = –2

Q.

An integrating factor of the differential equation $(1+y+x^{2}y)dx+(x+x^3)dy=0$ is

1. $\log x$

2. $x$

3. $e^x$

4. $\frac{1}{x}$

5. $-\frac{1}{x}$

Q. The solution for the differential equation $\frac{dy}{y}+\frac{dx}{x}=0$ is

1. $\log x.\log y=c$
2. $xy=c$
3. $\frac{1}{y}+\frac{1}{x}=c$
4. $x+y=c$

Q.

If $x=k^2$ and $y=k$ is the solution of the equation $x-5y+6=0$, then the sum of values of $k$ is

1. $5$

2. $2$

3. $3$

4. $4$

Q.

The general solution of the differential equation $\sqrt{1+x^2+y^2+x^2{y}^2}+xy\frac{dy}{dx}=0$(where $c$ is a constant of integration)

1. $\begin{array}{l}\sqrt{1+y^2}+\sqrt{1+x^2}=\frac{1}{2}lo{g}_e\left(\frac{\sqrt{1+x^2}-1}{\sqrt{1+x^2}+1}\right)+c\end{array}$

2. $\begin{array}{l}\sqrt{1+y^2}-\sqrt{1+x^2}=\frac{1}{2}lo{g}_e\left(\frac{\sqrt{1+x^2}-1}{\sqrt{1+x^2}+1}\right)+c\end{array}$

3. $\begin{array}{l}\sqrt{1+y^2}+\sqrt{1+x^2}=\frac{1}{2}lo{g}_e\left(\frac{\sqrt{1+x^2}+1}{\sqrt{1+x^2}-1}\right)+c\end{array}$

4. $\begin{array}{l}\sqrt{1+y^2}-\sqrt{1+x^2}=\frac{1}{2}lo{g}_e\left(\frac{\sqrt{1+x^2}+1}{\sqrt{1+x^2}-1}\right)+c\end{array}$

Q. The solution of the differential equation $y^{4}dx+2x{y}^{3}dy=\frac{ydx-xdy}{x^3{y}^3}$ is
(where $c$ is integration constant)

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

Q. The general solution of the differential equation $(y^2+e^{2x})dy-y^{3}dx=0$ (C being the constant of integration), is

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

Q. Which of the following is the solution of the DE $\frac{\mathrm{dy}}{\mathrm{dx}}=(3x+2y+4{)}^2$?

1. $\frac{1}{\sqrt{6}}{\tan }^{-1}\left(\sqrt{\frac{2}{3}}(3x+2y+4)\right)=x+c$
2. $\frac{1}{\sqrt{6}}{\tan }^{-1}\left(\sqrt{\frac{2}{3}}(3x+2y)\right)=x+c$
3. $\frac{1}{\sqrt{2}}{\tan }^{-1}\left(\sqrt{\frac{2}{3}}(3x+2y+4)\right)=x+c$
4. $\frac{1}{\sqrt{2}}{\tan }^{-1}\left(\sqrt{\frac{2}{3}}(3x+2y)\right)=x+c$

Q. The solution of $\frac{dy}{dx}+1=e^{x+y}$ is

1. $e^{-(x+y)}+x+c=0$
2. $e^{-(x+y)}-x+c=0$
3. $e^{x+y}+x+c=0$
4. $e^{x+y}-x+c=0$

Q. The general solution of the differential equation $(y^2–x^3)dx–xydy=0(x\ne 0)$ is : (where $c$ is a constant of integration)

1. $y^2+2x^2+c{x}^3=0$
2. $y^2-2x^3+c{x}^2=0$
3. $y^2+2x^3+c{x}^2=0$
4. $y^2–2x^2+c{x}^3=0$

Q.

For the given differential equation find the particular solution satisfying the given conditions.

(x+y)dy+(x-y)dx=0, y=1 when x=1.

Q.

Show that the given differential equation is homogeneous and then solve it.

$y^{\prime }=\frac{x+y}{x}$

Q.

Solve the equation –x² + x – 2 = 0

Q. A solution of the differential equation ${\left(\frac{dy}{dx}\right)}^2-x\frac{dy}{dx}+y=0$ is

1. y =2
2. y =2x
3. y =2x -4
4. $y=2x^2-4$

Q. The general solution of the differential equation $ydx-(x+y)dy=0$ is

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

Q. Solution of the differential equation $\frac{dy}{dx}+x(x+y)=x^3(x+y{)}^3-1$ is:
(where $C$ is integration constant)

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

Q.

The system of linear equations $\lambda x+2y+2z=5;2\lambda x+3y+5z=8;4x+\lambda y+6z=10$has:

1. no solution when $\lambda =2$

2. infinitely many solution when $\lambda =2$

3. no solution when $\lambda =8$

4. a unique solution when $\lambda =8$

Q. The solution of the differential equation
$x\frac{dy}{dx}+2y=x^2(x\ne 0)$ with $y\left(1\right)=1$ is

1. $y=\frac{3}{4}{x}^2+\frac{1}{4x^2}$
2. $y=\frac{4}{5}{x}^3+\frac{1}{5x^2}$
3. $y=\frac{x^2}{4}+\frac{3}{4x^2}$
4. $y=\frac{x^3}{5}+\frac{1}{5x^2}$

Q. What is the solution of the DE $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{yf}\left(x\right)-y^2}{f\left(x\right)}?$

1. $f\left(x\right)+y=c$
2. $f\left(x\right)=y(x+c)$
3. $y=\frac{c}{f\left(x\right)}+1$
4. $y=\mathrm{xf}\left(x\right)+c$

Q. The general solution of the differential equation $x{y}^2\frac{dy}{dx}=1-x^2+y^2-x^2{y}^2$ is (where $c$ is a constant of integration and log is considered with base ${}^{\prime }{e}^{\prime }$)

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

Q. The general solution of the differential equation $\frac{dy}{dx}=\frac{3x-4y+1}{4x+2y+3}$ is
(where $c$ is constant of integration)

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

Q. The slope of the tangent at (x, y) to a curve passing through $(1,\frac{\pi }{4})$ is given by $\frac{y}{x}-co{s}^2\left(\frac{y}{x}\right)$, then the equation of the curve is

1. $Noneofthese$
2. $y=ta{n}^{-1}\left[log\left(\frac{e}{x}\right)\right]$
   
3. $y=xta{n}^{-1}\left[log\left(\frac{x}{e}\right)\right]$
   
4. $y=xta{n}^{-1}\left[log\left(\frac{e}{x}\right)\right]$
   

Q.

For the given differential equation find the general solution.
$xlogx\frac{dy}{dx}+y=\frac{2}{x}logx$

Q. The general solution of the differential equation $xy(y+1)dy=(1+x^2)dx$ is
(where $c$ is constant of integration)

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

Q. The general solution of the differential equation $\frac{dy}{dx}+x(x+y)=x^3(x+y{)}^3-1$ is
(where ${}^{\prime }{C}^{\prime }$ is the constant of integration)

1. $\frac{1}{(x+y{)}^2}=C{e}^{x^2}+x^2+1$
2. $\frac{1}{(x+y{)}^2}=C{e}^x+x+1$
3. $\frac{1}{y^2}=C{e}^{x^2}+x^2+1$
4. $\frac{1}{(x+y)}=e^{x^2}+Cx+1$

Q. $\mathrm{The}\mathrm{solution}\mathrm{of}\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{x+y+1}{2x+2y+3}\mathrm{is}$.

1. $\frac{2}{3}\left(x+y\right)+\frac{1}{9}\log \left(2x+2y+4\right|=x+c$
2. $\frac{2}{3}\left(x+y\right)+\frac{1}{9}\log \left(3x+3y+4\right|=x+c$
3. $\frac{2}{3}\left(x+y\right)+\frac{1}{9}\log \left(3x+3y\right|=x+c$
4. $\frac{1}{9}\log \left(3x+3y+4\right|=x+c$

Q.

Solve the following system of equations and show all work.
$y=2x^2$
$y=–3x-1$