Question

The solution of $2x\frac{dy}{dx}+y=2x^3$ is

• A. $y\sqrt{x}=\frac{2}{5}{x}^{\frac{5}{2}}+c$
• B. $y\sqrt{x}=\frac{2}{7}{x}^{\frac{7}{2}}+c$
• C. $y\sqrt{x}=\frac{2}{9}{x}^{\frac{9}{2}}+c$
• D. $y\sqrt{x}=\frac{2}{3}{x}^{\frac{3}{2}}+c$

Answer 1

Answer: (A)

$y\sqrt{x}=\frac{2}{5}{x}^{\frac{5}{2}}+c$

$2x\frac{dy}{dx}+y=2x^3$

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

Here , $P=\frac{1}{2x}Q=x^2$

$I.F=e^{\int pdx}=e^{\int \frac{1}{2x}dx}=e^{\frac{1}{2}\log x}=\sqrt{x}$

solution of given D.E is

$y\times I.F=\int Q.I.Fdx+c$

$\Rightarrow y\times \sqrt{x}=\int x^2\times \sqrt{x}dx+c$

$\Rightarrow y\sqrt{x}=\int x^{\frac{5}{2}}dx+c$

$\Rightarrow y\sqrt{x}=\frac{2}{7}{x}^{\frac{7}{2}}+c$