Question

Solve the given differential equation by undetermined coefficients:

y'' + 4y = 4sin(2x)
y(x) =

Answer 1

Solving the differential equation involves finding the solution for the homogeneous part and guessing a particular solution for the non-homogeneous part, which is then adjusted to satisfy the equation. The calculated particular solution gets added to the general homogeneous solution.

Step-by-step explanation:

To solve the given differential equation by undetermined coefficients, we follow these steps:

1. First, we solve the corresponding homogeneous equation y'' + 4y = 0.
2. We then find the particular solution by guessing a form that can satisfy the non-homogeneous part of the equation, which in this case is 4sin(2x).
3. Finally, we derive parameters for our particular solution based on the non-homogeneous part and add it to the general solution of the homogeneous equation.

The homogeneous equation has solutions in the form of A cos(2x) + B sin(2x). For the non-homogeneous equation, a good guess for the particular solution is C x cos(2x) + D x sin(2x) because cos(2x) and sin(2x) are solutions to the homogeneous equation. Then we will find the values of C and D by plugging the guess into the original differential equation and solving for these constants.