Question

The expression satisfying the differential equation $(x^2-1)\frac{dy}{dx}+2xy=1$ is

• A. $a^{2}y-x{y}^2=c$
• B. $(y^2-1)x=y+c$
• C. $(x^2-1)y=x+c$
• D. none of these

Answer 1

Answer: (C)

Rewrite the given differential equation as follows :
$\frac{dy}{dx}+\frac{2x}{x^2-1}y=\frac{1}{x^2-1}$ which is a linear form
The integrating factor I.F.
$=e^{\int \frac{2x}{x^2-1}dx}=e^{\ell n\left(x^2-1\right)}=x^2-1$
Thus multiplying the given equation by $\left(x^2-1\right)$
we get $\left(x^2-1\right)\frac{dy}{dx}+2xy=1$
$\Rightarrow \frac{d}{dx}\left[y\left(x^2-1\right)\right]=1$
On integrating we get $y\left(x^2-1\right)=x+c$