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

General FormEdit

 , where   is a polynomial differential operator of degree   with constant coefficients.

General Form of the SolutionEdit

General solution is of the form  


  is called the complimentary solution, and is the solution of associated homogenous equation,  .

  is called the particular solution, obtained by solving  

Methods to find Complimentary SolutionEdit

Methods to solve for complimentary solution is discussed in detail in the article Second Order Homogeneous Ordinary Differential Equations.

Methods to find Particular SolutionEdit

Guessing method or method of undetermined coefficientsEdit

Choose appropriate y_p (x) with respect to g(x) from table below:


Find  , equate coefficients of terms and find the constants   and/or   and/or  . If it leads to an undeterminable situation, put   until it’s solvable.

Variation of parametersEdit

This method is applicable for inhomogeneous ODE with variable coefficients in one variable.

Suppose two linearly independent solutions of the ODE are known. Then


Solving by Laplace TransformsEdit

When initial conditions are given,

  1. Find Laplace Transform of either sides (See notes in earlier chapter for few common transforms)
  2. Isolate F(s)
  3. Split R.H.S. into partial fractions
  4. Find inverse Laplace Transforms.

Using ConvolutionsEdit

While solving by Laplace Transforms, if finally   is of the form </math>g(s)h(s)</math>, use property of convolutions that


and hence  .

Second Order Homogeneous Ordinary Differential Equations · About the Book