The general solution of the differential equation e^x dy + (y e^x + 2…

The correct answer is Option (3) → $y e^x+x^2= C$, where C is constant of integration

Given differential equation

$e^x dy+(ye^x+2x)dx=0$

Divide by $e^x$

$dy+\left(y+2xe^{-x}\right)dx=0$

Write in linear form

$\frac{dy}{dx}+y=-2xe^{-x}$

Integrating factor

$e^{\int 1dx}=e^x$

Multiply equation by $e^x$

$e^x\frac{dy}{dx}+ye^x=-2x$

$\frac{d}{dx}(ye^x)=-2x$

Integrate

$ye^x=-x^2+C$

$ye^x+x^2=C$

The general solution is $ye^x+x^2=C$.