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  4. The general solution of the differential equation y(x2y + ex)dx -exdy = 0, is

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Q. The general solution of the differential equation $y(x^2y + e^x)dx -e^xdy = 0$, is

Differential Equations

Solution:

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