Solve the initial-value problem using the method of undetermined coefficients. y"+16y= e^x + x^3.
y(0)=4,
y'(0)=0

The homogeneous part of the ODE has characteristic equation



with roots at , so the characteristic solution would be



As a guess for the solution to the nonhomogeneous part, we can try




Substituting into the ODE gives





so that the particular solution is



and the general solution to the ODE is




Given the initial values, we have




so that the solution to the IVP is