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