A) \[y+{{x}^{2}}+2x+2=c{{e}^{x}}\]
B) \[y+x+{{x}^{2}}+2=c{{e}^{2x}}\]
C) \[y+x+2{{x}^{2}}+2=c{{e}^{x}}\]
D) \[{{y}^{2}}+x+{{x}^{2}}+2=c{{e}^{x}}\]
Correct Answer: A
Solution :
\[y+{{x}^{2}}=\frac{dy}{dx}\] Þ \[\frac{dy}{dx}-y={{x}^{2}}\] This is the linear differential equation in y, where \[P=-1,\,Q={{x}^{2}}\] I.F. \[={{e}^{\int{P.dx}}}\]\[={{e}^{\int{-dx}}}={{e}^{-x}}\] Hence solution, \[y.\,(\text{I}\text{.F}).=\int{Q.(\text{I}\text{.F})\,dx+c}\] Þ \[y{{e}^{-x}}=-{{x}^{2}}{{e}^{-x}}-2x{{e}^{-x}}-2{{e}^{-x}}+c\] Þ \[y+{{x}^{2}}+2x+2=c{{e}^{x}}\].
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