Final answer:
To solve the differential equation by variation of parameters, follow these steps: find the complementary solution, find the particular solution using the method of variation of parameters, find u1' and u2' by solving a system of equations, substitute the values of u1' and u2' into the particular solution, and combine the solutions to get the general solution.
Step-by-step explanation:
To solve the differential equation by variation of parameters: 4y'' - y = ex/2 7
- First, find the complementary solution to the homogeneous equation by setting ex/2 7 equal to zero and solving for y.
- Next, find the particular solution using the method of variation of parameters. Assume the particular solution is of the form y = u1y1 + u2y2, where y1 and y2 are the solutions to the complementary solution, and u1' and u2' are functions to be determined.
- Find u1' and u2' by solving the system of equations (y1u1' + y2u2')' = ex/2 7 and (y1u1' + y2u2')' = 0.
- Substitute the values of u1' and u2' into the particular solution y = u1y1 + u2y2.
- Combine the complementary and particular solutions to get the general solution to the differential equation.