Solution of Differential Equation d^2y/dx^2 = 4y = tan 2x

First Order Linear Differential Equation

The given differential equation is a second-order linear differential equation. We can convert this into a first-order linear differential equation by introducing a new variable v = dy/dx.

Substituting for v, we get dv/dx = d^2y/dx^2 = 4y - tan 2x.

Hence, the first-order linear differential equation is dv/dx - 4y = tan 2x.

Homogeneous Solution

The homogeneous solution of the differential equation is obtained by setting tan 2x equal to zero. Hence, the homogeneous solution is yh = c1e^(2x) + c2e^(-2x), where c1 and c2 are constants.

Particular Solution

To find the particular solution, we need to find a function yp such that dv/dx - 4yp = tan 2x.

We can guess a particular solution of the form yp = a tan 2x + b, where a and b are constants.

Differentiating yp with respect to x, we get dy/dx = 2a sec^2 2x.

Differentiating again, we get d^2y/dx^2 = 8a tan 2x sec^2 2x.

Substituting for dy/dx and d^2y/dx^2 in the first-order linear differential equation, we get 8a tan 2x sec^2 2x - 4(a tan 2x + b) = tan 2x.

Simplifying, we get 4a tan 2x - 4b = 0.

Hence, a = b = 1/4.

Therefore, the particular solution is yp = (1/4) tan 2x + 1/4.

General Solution

The general solution of the differential equation is y = yh + yp = c1e^(2x) + c2e^(-2x) + (1/4) tan 2x + 1/4.

Final Answer

The solution of the differential equation d^2y/dx^2 = 4y - tan 2x is y = c1e^(2x) + c2e^(-2x) + (1/4) tan 2x + 1/4.