Question

Solve: $x(x-1)\frac{dy}{dx}-(x-2)y=x^3(2x-1)$

Answer 1

Given,

$x(x-1)\frac{dy}{dx}-(x-2)y=x^3(2x-1)$

$\frac{dy}{dx}-\frac{(x-2)}{x(x-1)}y=\frac{x^2(2x-1)}{(x-1)}$

$\frac{dy}{dx}+Py=Q$

$P=-\frac{(x-2)}{x(x-1)}\to I.F=e^{\int Pdx}=e^{\int -\frac{(x-2)}{x(x-1)}dx}=e^{\log \frac{x-1}{x^2}}=\frac{x-1}{x^2}$

$Q=\frac{x^2(2x-1)}{(x-1)}$

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

$y\times \left(\frac{x-1}{x^2}\right)=\int \frac{x^2(2x-1)}{(x-1)}\times \frac{x-1}{x^2}dx$

$y\left(\frac{x-1}{x^2}\right)=\int (2x-1)dx$

$y\left(\frac{x-1}{x^2}\right)=x(x-1)+c$

$y=x^3+C$