Final answer:
The solution to the differential equation y'' - 4y = sin³(x) involves finding the homogeneous solution yh, and a particular solution yp using the method of undetermined coefficients after simplifying sin³(x) with a trigonometric identity.
Step-by-step explanation:
We are asked to solve the differential equation y'' - 4y = sin³(x). This is a non-homogeneous linear differential equation where the non-homogeneous part is sin³(x). We first handle the homogeneous part, y'' - 4y = 0, which has the solution of the form yh = C1e2x + C2e-2x where C1 and C2 are constants determined by initial conditions.
To find the particular solution to the non-homogeneous part, we use the method of undetermined coefficients. However, since sin³(x) is not a simple trigonometric function, we first need to apply a trigonometric identity to simplify it: sin³(x) = (3sin(x) - sin(3x))/4. Hence, our particular solution will be of the form yp = A sin(x) + B sin(3x). We would then substitute yp into the differential equation to solve for A and B.
Once we have the constants A and B, our general solution will be the sum of the homogeneous and particular solutions: y(x) = yh(x) + yp(x).