The general solution of the differential equation \(e^{x}dy+(ye^{x}+2x)dx=0\) is
The general solution of the differential equation \(e^{x}dy+(ye^{x}+2x)dx=0\) is
\(xe^{y}+x^{2}=C\)
\(xe^{y}+y^{2}=C\)
\(ye^{x}+x^{2}=C\)
\(ye^{y}+x^{2}=C\)
The Correct Option is C
Solution and Explanation
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.
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