Final answer:
To solve the initial-value problem, find the complementary function for the homogeneous equation, then find a particular solution for the nonhomogeneous equation, and finally, apply the initial conditions to find the specific solution.
Step-by-step explanation:
The student is asked to solve an initial-value problem involving a second-order linear differential equation with non-constant coefficients. The initial conditions given are y(0)=0 and y'(0)=0. To solve this, we first find the complementary function (yc) by solving the homogeneous equation 2y" + 3y' - 2y = 0. Then we find a particular solution (yp) to the nonhomogeneous equation.
1. Solve the characteristic equation associated with the homogeneous part: r2 + (3/2)r - 1 = 0.
2. Find the roots of the characteristic equation and construct the complementary solution yc(x).
3. Use a method such as undetermined coefficients or variation of parameters to find yp(x) that satisfies the nonhomogeneous equation.
4. Combine the complementary solution yc and the particular solution yp to get the general solution y(x).
5. Apply the initial conditions to solve for any constants in the general solution and obtain the specific solution to the initial-value problem.