Question

The solution of the differential equation $ x\frac{dy}{dx}+2y={{x}^{2}} $ is:

• A. $ y=\frac{{{x}^{2}}+c}{4{{x}^{2}}} $
• B. $ y=\frac{{{x}^{2}}}{4}+c $
• C. $ y=\frac{{{x}^{4}}+c}{{{x}^{2}}} $
• D. $ y=\frac{{{x}^{4}}+c}{4{{x}^{2}}} $
• E. $ y=\frac{{{x}^{3}}}{4}+\frac{c}{{{x}^{2}}} $

Answer 1

Answer: (D)

$ x=\frac{dy}{dx}+2y={{x}^{2}} $
On differentiating w.r.t $ x, $
we get $ \frac{dy}{dx}+\frac{2}{x}y=x $ Integrating factor
$={{e}^{\int{\frac{2}{x}dx}}}={{x}^{2}} $
$ \therefore $ Required solution is $ y.{{x}^{2}}=\int{{{x}^{3}}}dx=\frac{{{x}^{4}}}{4}+c=\frac{{{x}^{4}}+c}{4} $
$ \therefore $ $ y=\frac{{{x}^{4}}+c}{4{{x}^{2}}} $