Solving Second Order Ordinary Differential Equations
Homogenous
Consider the function
By applying the function to the above equation, we get:
The above equation is called the Auxiliary equation or Characteristic equation.
Case 1: Distinct real roots
Case 2: Equal real roots
Case 3: Complex conjugate roots
Non-homogenous
Method of undetermined coefficients
We find
If
- a constant,
is a constant , , or , , (Only works if is not a root of auxiliary equation)
Steps
- Solve for
- Based on the form of
, make an initial guess for . - Check if any term in the guess for
is a solution to the complementary equation. - If so, multiply the guess by
. Repeat this step until there are no terms in that solve the complementary equation. - Substitute
into the differential equation and equate like terms to find values for the unknown coefficients in . - If coefficients were unable to be found (they cancelled out or something like
that), multiply the guess by
and start again.