Question

Solve the differential equation is $x(x-1)\frac{dy}{dx}-(x-2)y=x^3(2x-1)$.

• A. $y(x-1)=x^2(x^2-x+c)$
• B. $y(x+1)=x^2(x^2+x+c)$
• C. $y(x+1)=x^2(x^2-x+c)$
• D. $y(x-1)=x^2(x^2+x+c)$

Answer 1

The correct option is A $y(x-1)=x^2(x^2-x+c)$

$x(x-1)\frac{dy}{dx}-(x-2)y=x^3(2x-1)\Rightarrow \frac{dy}{dx}-\frac{x-2}{x(x-1)}y=\frac{x^2(2x-1)}{(x-1)}$ ...(1)
Here $P=-\frac{x-2}{x(x-1)}\Rightarrow \int Pdx=-\int \frac{x-2}{x(x-1)}dx=-\int (\frac{2}{x}-\frac{1}{x-1})dx$
$=\log (1-x)-2\log x=\log \left(\frac{1-x}{x^2}\right)$
$\therefore I.F.=e^{\log \left(\frac{1-x}{x^2}\right)}=\frac{1-x}{x^2}$
Multiplying (1) by $I.F.$, we get
$\frac{1-x}{x^2}\frac{dy}{dx}-\frac{x-2}{x^3}y=(2x-1)$
Integrating both sides we get
$\frac{1-x}{x^2}y=\int (2x-1)dx+c=x^2-x+c\Rightarrow y(1-x)=x^2(x^2-x+c)$