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

Correct Answer: 97

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 \( m + n = 97 \).