Given that $y = 2$ at $x = \frac{\pi}{4}$, solve the differential equation $$\frac{dy}{dx} = \frac{3}{y \cos x}$$ - Edexcel - A-Level Maths Pure - Question 6 - 2012 - Paper 7

At $x = \frac{\pi}{4}$, $y = 2$. We substitute these values into the equation:

$\frac{1}{2} (2)^2 = 3 \tan \left( \frac{\pi}{4} \right) + C$

This simplifies to:

$2 = 3(1) + C$

Thus,

$C = 2 - 3 = -1$

Substituting back the value of $C$ into the integrated equation, we have:

$\frac{1}{2} y^2 = 3 \tan x - 1$

To write this in a standard form:

$y^2 = 6 \tan x - 2$

This is the solution to the differential equation.