Ordinary Differential Equations:Cheat Sheet/Second Order Homogeneous Ordinary Differential Equations

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With Constant Coefficients[edit]

General Form[edit]

or , where

is called the polynomial differential operator with constant coefficients.


  1. Solve the auxiliary equation, , to get
  2. If are
    1. Real and distinct, then
    2. Real and equal, then
    3. Imaginary, , then

Euler-Cauchy Equations[edit]

General Form[edit]

or where

is called the polynomial differential operator.


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]
  1. If are linearly dependent,
  2. If , for some , then .

First Order Ordinary Differential Equations · Second Order Inhomogeneous Ordinary Differential Equations