Question

The differential equation $x\frac{dy}{dx}-y=x^2$, has the general solution

• A. $y-x^2=cx$
• B. $2y-x^3=cx$
• C. $2y+x^2=2cx$
• D. $y+x^2=2cx$

Answer 1

The correct option is A $y-x^2=cx$

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

$\Rightarrow \frac{dy}{dx}-\frac{y}{x}=\frac{x^2}{y^2}$

$\frac{dy}{dx}-\left(\frac{1}{x}\right)y=x$

$\because \frac{dy}{dx}+py=Q$

$p=\frac{-1}{x},Q=x$

If $e^{\int pdx}=e^{\int -\frac{1}{x}dx}=e^{-logx}$

$=e^{logx-1}=\frac{1}{x}$

$\therefore$ General $so{l}^n$

$y\left(\frac{1}{x}\right)=\int Q\left(IF\right)dx+c$

$\frac{y}{x}=\int x.\frac{1}{x}dx+c$

$\frac{y}{x}=\int 1dx+c$

$\frac{y}{x}=x+c$

[$y=x^2+cx]$