Question

Let $y = y(x)$ be the solution of the differential equation $(1 - x^2) \, dy = \left[ xy + \left( x^3 + 2 \right) \sqrt{3 \left( 1 - x^2 \right)} \right] dx$, $-1 < x < 1, y(0) = 0$. If $y\left( \frac{1}{2} \right) = \frac{m}{n}$, $m$ and $n$ are coprime numbers, then $m + n$ is equal to \\_\\_\\_\\_\\_\\_\\_\\_\\_.

Answer 1

Solution: Rewrite the differential equation and solve by separation of variables:

$\frac{dy}{dx} = \frac{xy + \left(x^3 + 2\right)\sqrt{1 - x^2}}{1 - x^2}$

Using integration factors and simplifying, we obtain:

$y = \sqrt{3} \left(\frac{65}{32}\right)$

where $m = 65$ and $n = 32$, giving $m + n = 97$.