Solving Homogeneous Differential Equations

Q.

Solution of the differential equation $\frac{\sqrt{x}dx+\sqrt{y}dy}{\sqrt{x}dx-\sqrt{y}dy}=\sqrt{\frac{y^3}{x^3}}$ is given by

1. $\frac{3}{2}log\left(\frac{y}{x}\right)+log\left|\frac{x^{3/2}+y^{3/2}}{x^{3/2}}\right|+ta{n}^{-1}{\left(\frac{y}{x}\right)}^{3/2}+c=0$

2. $\frac{2}{3}log\left(\frac{y}{x}\right)+log\left|\frac{x^{3/2}+y^{3/2}}{x^{3/2}}\right|+ta{n}^{-1}\frac{y}{x}+c=0$

3. $\frac{2}{3}log\left(\frac{y}{x}\right)+log\left(\frac{x+y}{x}\right)+ta{n}^{-1}\left(\frac{y^{3/2}}{x^{3/2}}\right)+c=0$

4. None of the above

Q.

The equation of the normal of the curve $\mathrm{y}={\left(1+\mathrm{x}\right)}^{\mathrm{y}}+{\sin }^{-1}\left({\sin }^2\mathrm{x}\right)$ at $\mathrm{x}=0$ is

1. $\mathrm{x}+\mathrm{y}=1$

2. $\mathrm{x}-\mathrm{y}=1$

3. $\mathrm{x}+\mathrm{y}=-1$

4. $\mathrm{x}-\mathrm{y}=-1$

Q. An equation of the curve satisfying $xdy-ydx=\sqrt{x^2-y^2}$ dx and $y\left(1\right)=0$ is

1. $y=x^{2}log|sinx|$
2. $y=xsin\left(log|x|\right)$
3. $y^2=x(x-1{)}^2$
4. $y=2x^2(x-1)$

Q. The family of curves which satisfies the differential equation $y{e}^{\frac{x}{y}}dx=(x{e}^{\frac{x}{y}}+y^2)dy,(y\ne 0)$ is
(where ${}^{\prime }{C}^{\prime }$ is the constant of integration )

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

Q.

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

[EAMCET 2003]

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. The particular solution of the differential equation $x\frac{dy}{dx}=y\ln \left(\frac{y}{x}\right),x\ne 0$ is $y=f\left(x\right)$. If $f\left(1\right)=e^2$, then

1. $f\left(x\right)=x{e}^{1+x^2}$
2. $f\left(x\right)=x{e}^{1+x}$
3. there are more than one tangent parallel to $x-$axis to the curve $y=f\left(x\right)$
4. there is exactly one tangent parallel to $x-$axis to the curve $y=f\left(x\right)$

Q.

What is the value of $t$ in the equation $\frac{10}{t}=1$?

1. $10$

2. $1$

3. $0$

4. $\frac{1}{10}$

Q. Solution of the differential equation : $\frac{dy}{dx}=\frac{3x^2{y}^4+2xy}{x^2-2x^3{y}^3}$ is

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

Q. The solution of $x^2\frac{dy}{dx}-xy=1+\cos \frac{y}{x}$ is

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

Q. The curve amongst the family of curves represented by the differential equation, $(x^2-y^2)dx+2xydy=0$ which passes through $(1,1)$, is :

1. a circle with centre on the x-axis
2. a circle with centre on the y-axis
3. an ellipse with major axis along the y-axis
4. a hyperbola with transverse axis along the x-axis

Q. The solution of $(y+x+5)dy=(y-x+1)dx$ is

1. $log\left(\right(y+3{)}^2+(x+2{)}^2)+ta{n}^{-1}\frac{y+3}{x+2}=C$
2. $log\left(\right(y+3{)}^2+(x-2{)}^2)+ta{n}^{-1}\frac{y-3}{x-2}=C$
3. $log\left(\right(y+3{)}^2+(x+2{)}^2)+2ta{n}^{-1}\frac{y+3}{x+2}=C$
4. $log\left(\right(y+3{)}^2+(x+2{)}^2)-2ta{n}^{-1}\frac{y+3}{x+2}=C$

Q. An equation of the curve satisfying $xdy-ydx=\sqrt{x^2-y^2}$ dx and $y\left(1\right)=0$ is

1. $y=x^{2}log|sinx|$
2. $y=xsin\left(log|x|\right)$
3. $y^2=x(x-1{)}^2$
4. $y=2x^2(x-1)$

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. The solution of $(y+x+5)dy=(y-x+1)dx$ is

1. $log\left(\right(y+3{)}^2+(x+2{)}^2)+ta{n}^{-1}\frac{y+3}{x+2}=C$
2. $log\left(\right(y+3{)}^2+(x-2{)}^2)+ta{n}^{-1}\frac{y-3}{x-2}=C$
3. $log\left(\right(y+3{)}^2+(x+2{)}^2)+2ta{n}^{-1}\frac{y+3}{x+2}=C$
4. $log\left(\right(y+3{)}^2+(x+2{)}^2)-2ta{n}^{-1}\frac{y+3}{x+2}=C$

Q. If $x\frac{dy}{dx}=y(logy-logx+1)$, then the solution of the equation is

1. $ylog\left(\frac{x}{y}\right)=cx$
2. $xlog\left(\frac{y}{x}\right)=cy$
3. $log\left(\frac{y}{x}\right)=cx$
4. $log\left(\frac{x}{y}\right)=cy$