Final answer:
To solve the nonhomogeneous differential equation by variation of parameters, find the complementary solution to the homogeneous equation, then determine particular solutions to form the general solution.
Step-by-step explanation:
To solve the given differential equation y" + 3y' + 2y = 17ex by variation of parameters, we first find the complementary solution to the homogeneous equation y" + 3y' + 2y = 0. The characteristic equation of this homogeneous equation is r2 + 3r + 2 = 0, which factors as (r + 1)(r + 2) = 0. Thus, we get two roots r=-1 and r=-2. The complementary solution (yc) is yc = C1e-x + C2e-2x, where C1 and C2 are constants.
Next, we apply the variation of parameters technique, which involves finding particular solutions u1(x) and u2(x) such that yp = u1(x)e-x + u2(x)e-2x is a solution to the nonhomogeneous equation. To find u1 and u2, we set up a system of equations based on the derivatives of yp and then solve for u1 and u2.
After obtaining u1 and u2, we can express the general solution y(x) to the original differential equation as y(x) = yc + yp. The particular solution yp involves integrating the expressions for u1 and u2 and substituting back into yp. Finally, we determine any unknown constants by applying initial conditions, if they are given.