Euler substitution

The Euler substitution is a useful substitution for solving linear homogeneous ordinary differential equations with constant coefficients. These are differential equations of the form: The solution will be a linear combination of n linearly independent eigenfunctions, : The Euler substitution allows one to determine these eigenfunctions. It simplifies the problem as instead of having to solve a differential equation, one must instead solve a polynomial

Method

Consider solving a general equation of the form above: The Euler substitution is . This yields, for the first few derivatives: Substituting back in, this yields: Dividing through by , This is a polynomial and very easy to solve. If it has solutions λi, the solution to the differential equation can be written as: Where the ci are arbitrary constants. If a root is repeated, then there will not be n linearly independent solutions. In this case, if the root is repeated m times, then m linearly independent functions can be created by multiplying by 1, t, t2...tm-1.

In the important case of n=2, a second order equation, this polynomial can be written as . This equation can easily be solved using the quadratic formula. There are three cases that arise:

Case I: When In this case, the solutions are exponentials: Case II: When In this case the solutions are complex exponentials. These can be rewritten in terms of sines and cosines using Euler's formula: and , When Case III: In this case there is only a single solution for λ as it is a repeated root. Therefore is the same as and so are not linearly independent. Using the above, m=2 and so we can multiply by 1 and t. Therefore, for this case the general solution is: A particular solution can then be found by applying boundary conditions.

Example

Consider the differential equation: Substituting in in and then dividing by produces: This only has one one solution, namely λ=1. This is a repeated root, so we multiply one solution by t to form 2 linearly independent solutions. Thus the general solution is: 