Question

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

• A. $y=x^3+\frac{x^2}{x-1}+c$
• B. $y=x^3+\frac{c{x}^3}{x-1}$
• C. $y=x^3+\frac{c{x}^2}{x-1}$
• D. None of these.

Answer 1

The correct option is C $y=x^3+\frac{c{x}^2}{x-1}$

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

$\Rightarrow y=x^3+\frac{c{x}^2}{x-1}$