Question

Let $y=y(x),x>1$, be the solution of the differential equation $(x-1)\frac{dy}{dx}+2xy=\frac{1}{x-1}$, with $y(2)=\frac{1+e^4}{2e^4}$. If $y(3)=\frac{e^{\alpha }+1}{\beta e^{\alpha }}$. then the value of $\alpha +\beta$ is equal to______.

Answer 1

Answer: 14

$\frac{dy}{dx}+\frac{2x}{x-1}\cdot y=\frac{1}{(x-1{)}^2}$
$y=\frac{1}{(x-1{)}^2}\left[\frac{e^{2x}+1}{2e^{2x}}\right]$
$y(3)=\frac{e^6+1}{8e^6}$
$\alpha +\beta =14$