Question

If $y(x) = C_1 \cos 2x + C_2 \sin 2x + A(x) \log (\sec 2x + \tan 2x)$ is the general solution of the differential equation $\frac{d^2y}{dx^2} + 4y = \tan 2x$, then $A(x) = \_\_\_\_\_\_\_$}

• A. $-\frac{\sin 2x}{4}$
• B. $\frac{\sin 2x}{4}$
• C. $-\frac{\cos 2x}{4}$
• D. $\frac{\cos 2x}{4}$

Hint:

Using Variation of Parameters: the Wronskian of $\cos ax$ and $\sin ax$ is $a$.
The integration of the $u(x)$ term always leads to a $-\frac{1}{2a^2}\cos ax \log(\sec ax + \tan ax)$ component.
This directly yields the denominator of $2a^2 = 2(2^2) = 8$ or relative factor combinations matching option (C).

Answer 1

The Correct Option is C