Final answer:
To solve the initial value problem, we find the complementary solution, determine the particular solution, and use the initial conditions to find the constants. The result is a full equation for the motion based on position x.
Step-by-step explanation:
To solve the given second-order non-homogeneous linear differential equation with initial conditions, we'll first find the complementary solution (yc) by solving the associated homogeneous equation y'' + 4y = 0. Next, we'll determine a particular solution (yp) to the non-homogeneous equation. Finally, we construct the general solution, apply the initial conditions, and find the constants for the particular solution.
Complementary solution (yc):
- We solve the characteristic equation r2 + 4 = 0, finding r = ±2i.
- Thus, yc(x) = C1cos(2x) + C2sin(2x).
Particular solution (yp): Since the non-homogeneous part consists of sin(2x) and cos(2x), we guess yp = Axsin(2x) + Bxcos(2x) to avoid a solution similar to the complementary solution.
Substitute yp into the non-homogeneous differential equation to find A and B.
Applying initial conditions:
- Use y(0) = 0 to find C1 and B.
- Use y'(0) = 1 to find C2 and A.
After finding the constants A, B, C1, and C2, we construct the complete solution y(x) = yc(x) + yp(x). This results in a specific equation representing the motion as a function of position x.