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

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General Form[edit]

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

General Form of the Solution[edit]

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 Solution[edit]

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

Methods to find Particular Solution[edit]

Guessing method or method of undetermined coefficients[edit]

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 parameters[edit]

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 Transforms[edit]

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 Convolutions[edit]

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

and hence .

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