Ordinary Differential Equations:Cheat Sheet/Second Order Homogeneous Ordinary Differential Equations
With Constant Coefficients edit
General Form edit
or , where
- is called the polynomial differential operator with constant coefficients.
Solution edit
- Solve the auxiliary equation, , to get
- If are
- Real and distinct, then
- Real and equal, then
- Imaginary, , then
Euler-Cauchy Equations edit
General Form edit
or where
- is called the polynomial differential operator.
Solution edit
Solving is equivalent to solving
General Homogenous ODE with Variable Coefficients edit
If one particular solution is known edit
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 .
Abel's identity edit
For the homogeneous linear ODE , Wronskian of its two solutions is given by
Solution with Abel's identity edit
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 edit
- If are linearly dependent,
- If , for some , then .