y+x²=dy/dx has the solution

Correct Answer is : y+x2+2x+2=cexy+x²+2x+2=c e^xy+x2+2x+2=cex

Correct Answer is : y+x2+2x+2=cexy+x²+2x+2=c e^xy+x2+2x+2=cex

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EAMCET Engineering 2002 Solved Paper

Section: Mathematics

Question : 159 of 200

Marks: +1, -0

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

Solution:

We have,

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

\Rightarrow \frac{dy}{dx}-y=x^2

It is a linear differential equation. On comparing with

\frac{dy}{dx}+P y=Q

, we get

P=-1,\; Q=x^2

\text{IF}=e^{\int P\,dx}=e^{\int -1\,dx}=e^{-x}

Required solution is

y\cdot e^{-x}=\int x^2 e^{-x}\,dx+c

\Rightarrow y e^{-x}=-x^2 e^{-x}+2\int x e^{-x}\,dx

\Rightarrow y e^{-x}=-x^2 e^{-x}-2x e^{-x}+2\int e^{-x}\,dx

\Rightarrow y e^{-x}=-x^2 e^{-x}-2x e^{-x}-2e^{-x}+c

\Rightarrow y=-(x^2+2x+2)+c e^x

\Rightarrow y+x^2+2x+2=c e^x

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