Question

Solve $x\frac{dy}{dx}-ay=x^2+\frac{1}{2},x>0,a\ne 2,-1$

Answer 1

$x\frac{dy}{dx}-ay=x^2+\frac{1}{2}$

$\frac{dy}{dx}-\frac{ay}{x}=x+\frac{1}{2x}$

compare with

$\frac{dy}{dx}+p\left(x\right)y=q\left(x\right)$

No integrating factor

I.F$=e^{\int pdx}$

$=e^{a\int \frac{dx}{x}}$

$=e^{alogx}$

$=e^{lo{g}_e{x}^a}$

$=x^a$

So solution is

$y(T.F)=\int \theta I.F.dx+c$

$x^{a}y=\int (x+\frac{1}{2x})x^{a}dx+c$

$=\int x^{a+1}dx+\frac{1}{2}\int x^{a-1}dx+c$

$x^{a}y=\frac{x^{a+2}}{a+2}+\frac{1}{2}\frac{x^a}{a}+c$.