Question

Solve the differential equation $\frac{dy}{dx}+\frac{y}{x}=x^2$

• A. $x=\frac{x^4}{4}+C$
• B. $y=\frac{x^4}{4}+C$
• C. $xy=\frac{x^4}{4}+C$
• D. None of these

Answer 1

The correct option is C $xy=\frac{x^4}{4}+C$

# $\frac{dy}{dx}+\frac{y}{x}=x^2$ this linear differential equation

So: $p\left(x\right)=\frac{1}{x}$ $a\left(x\right)=x^2$

$I.F=e^{\int pdx}=e^{\int 1/xdx}=e^{lnx}=x$

$y.I.F=\int Q$. If $dx\Rightarrow y.x=\int x^2.xdx\Rightarrow$

$xy=\frac{x^4}{4}+c$