Missed this problem on an exam and im not sure how to get to

Missed this problem on an exam and im not sure how to get to the answer if it comes up again at some point.

Solution

Given that

y = ( c₁ + c₂x)cos 2x + ( c₃ + c₄x)sin 2x

We know that

For a linear differential equation with constant coefficients , if two pairs of roots(±i ,±i ) are equal and imaginary then the solution is y = e^x [ ( c₁ + c₂x)cos x + ( c₃ + c₄x)sin x ]

Hence ,

The given function is a general solution to the linear differential equation with constant coefficients