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

The Correct Option is C

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.