Crazy diffy! 3

$\large \frac { da }{ db } =a^2+\int _0^\frac \pi 4 a^2 \ db$

For the differential equation above, $a=f(b)$ and $a(0)=0$ . Find $\left\lceil a\left( 0.5 \right) \right\rceil$ .

Notes:

• You may use a calculator to do some heavy calculations.
• The problem is original