Higher Order ODEs

There are two sides to solving linear ODEs with constant coefficients: finding the complementary solution $y_c$ and a particular solution $y_p$. Because it is linear, adding them will result in all possible solutions; $y = y_c + y_p$.

Solving

Most problems will consist of being given an equation (we will assume it is nonhomogeneous) and some initial conditions.

1. Find the complementary solution.
   1. Make the equation homogeneous by setting $g(t)=0$.
   2. Create the characteristic equation.
   3. Find the solutions of $r$. It is usually most clear to factorize the characteristic equation as best as possible. Remember the difference of squares rule.
   4. From the solutions of $r$, create the general form for the complementary solution.
   5. You will have to wait until after you get the particular solution to find the coefficients.
2. Find a particular solution.
   1. Ansatz a form for the particular solution using the method of undetermined coefficients. Remember that if you find the same term in the Ansatz as in the complementary solution, you should multiply the term by $t$.
   2. Substitute the particular solution into the original equation.
   3. Use the equation to solve for the coefficients. DO NOT use the initial conditions provided to find the particular solution coefficients.
3. Add the complementary solution and particular solution to get the general solution.
4. Solve for the coefficients from the complementary solution.