Question

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

• A. $xe^{y}+x^{2}=C$
• B. $xe^{y}+y^{2}=C$
• C. $ye^{x}+x^{2}=C$
• D. $ye^{y}+x^{2}=C$

Answer 1

Answer: (C)

$ye^{x}+x^{2}=C$

Answer 2

The given differential equation is:

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

$⇒e^{x}\frac{dy}{dx}=ye^{x}+2x=0$

$⇒\frac{dy}{dx}+y=-2xe^{-x}$

This is a linear differential equation of the form

$\frac{dy}{dx}+py=Q,where\; p=1\; and\; Q=-2xe^{-x.}$

$Now,I.F.=e^{\int{pdx}}=e^{\int{dx}}=e^{x}.$

The general solution of the given differential equation is given by,

$y(I.F.)=\int{(Q×I.F.)dx}+C$

$⇒ye^{x}=\int{(-2xe^{-x}.e^{x})d}x+C$

$⇒ye^{x}=-\int{2xdx}+C$

$⇒ye^{x}=-x^{2}+C$

$⇒ye^{x}+x^{2}=C$

Hence,the correct answer is C.