Solving Homogeneous Differential Equations

Q.

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

[EAMCET 2003]

1. 

2. 
   

3. 
   

4. 
   

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. $y=ta{n}^{-1}\left(log\left(\frac{e}{x}\right)\right)$
2. $y=xta{n}^{-1}\left(log\left(\frac{x}{e}\right)\right)$
3. None of these
4. $y=xta{n}^{-1}\left(log\left(\frac{e}{x}\right)\right)$

Q. The solution of the differential equation $\frac{dy}{dx}=\frac{y}{x}+\frac{\varphi \left(\frac{y}{x}\right)}{\varphi \left(\frac{y}{x}\right)}$ is

1. $\varphi \left(\frac{y}{x}\right)=kx$
   
2. $\varphi \left(\frac{y}{x}\right)=ky$
   
3. $y\varphi \left(\frac{y}{x}\right)=k$
4. $x\varphi \left(\frac{y}{x}\right)=k$
   

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. The solution of $\frac{dy}{dx}=\frac{y}{x}+tan\frac{y}{x}$ is

1. $ysin\left(\frac{y}{x}\right)=cx$
2. $sin\left(\frac{x}{y}\right)=cx$
3. $sin\left(\frac{y}{x}\right)=cx$
4. $ysin\left(\frac{y}{x}\right)=cy$

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. Solution of the equation $xdy=(y+x\frac{f\left(\frac{y}{x}\right)}{f^{\prime }\left(\frac{y}{x}\right)})dx$

1. $f\left(\frac{x}{y}\right)=cy$
2. $f\left(\frac{y}{x}\right)=cx$
3. $f\left(\frac{y}{x}\right)=cxy$
4. $f\left(\frac{y}{x}\right)=0$

Q. Solution of the differential equation: $\frac{x+y\frac{dy}{dx}}{y-x\frac{dy}{dx}}=\frac{xsi{n}^2(x^2+y^2)}{y^3}$

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

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

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