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

With Constant CoefficientsEdit

General FormEdit

  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 EquationsEdit

General FormEdit

  or   where

  is called the polynomial differential operator.


Solving   is equivalent to solving  

General Homogenous ODE with Variable CoefficientsEdit

If one particular solution is knownEdit

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 identityEdit

For the homogeneous linear ODE  , Wronskian of its two solutions is given by  

Solution with Abel's identityEdit

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

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