Question

The general solution of the differential equation $y(x^{2}y+e^x)dx-e^{x}dy=0$, is

• A. $x^{3}y-3e^x=Cy$
• B. $x^{3}y+3e^x=3Cy$
• C. $y^{3}x-3e^y=Cx$
• D. $y^{3}x+3e^y=Cx$

Answer 1

Answer: (B)

Given, $y(x^{2}y+e^x)dx-e^{x}dy=0$
$\Rightarrow x^2{y}^{2}dx+y{e}^{x}dx-e^{x}dy=0$
$\Rightarrow x^{2}dx+\frac{y{e}^{x}dx-e^{x}dy}{y^2}=0$
$\Rightarrow x^{2}dx+d\left(\frac{e^x}{y}\right)=0$
$\Rightarrow \frac{x^3}{3}+\frac{e^x}{y}=C$
$\Rightarrow x^{3}y+3e^x=3Cy$