4.) Find the general solution to the following ODE and check the result by substitution.
The homogeneous characteristic equation for linear 2nd order ODE's with constant coefficients is
For the ODE in this problem, the characteristic equation becomes
To solve for the solutions, the discriminant to the quadratic equation must first be calculated.
Since the discriminant is negative, there will be two imaginary solutions to the ODE. The solutions can be obtained by solving the remainder of the quadratic formula.
The two distinct, linearly independent, homogeneous solutions for the case with two imaginary roots are
The homogeneous solution is
The final general solution is therefore
To check this result using substitution, we must take the first and second derivatives of the general solution to substitute back into the original ODE.
Substituting back into the original ODE gives
Since all terms cancel to 0, is a general solution to the original ODE.
Section 2 Lecture Notes