With Constant Coefficients
or , where
- is called the polynomial differential operator with constant coefficients.
- Solve the auxiliary equation, , to get
- If are
- Real and distinct, then
- Real and equal, then
- Imaginary, , then
- is called the polynomial differential operator.
Solving is equivalent to solving
General Homogenous ODE with Variable Coefficients
If one particular solution is known
If one solution of a homogeneous linear second order equation is known, , original equation can be converted to a linear first order equation using substitutions and subsequent replacement .
For the homogeneous linear ODE , Wronskian of its two solutions is given by
Solution with Abel's identity
Given a homogenous linear ODE and a solution of ODE, , find Wronskian using Abel’s identity and by definition of Wronskian, equate and solve for .
Few Useful Notes
- If are linearly dependent,
- If , for some , then .