Method of Undetermined Coefficients

Lesson 6W

The method of undetermined coefficients lets you solve non-homogeneous linear differential equations.

The Particular Solution

One key part of finding the general solution is to find one single solution. This is often called the particular solution.

The general solution

The other key part of finding the general solution is to find the general solution to the homogeneous version of the differential equation. Just take whatever is on the right hand side of the differential equation and change it to zero, so that the equation is homogeneous. Then solve that, and use the solution to build the general solution of the non-homogeneous equation.

Still to learn

In these videos, there are two important things I don’t discuss, but we will talk about soon. First: you need to know how to determine the form of the particular solution. Second: you need to know about “bumping up”.

Practice problem

Find the general solution for this differential equation: \[ y''-4y = 3t^3+t-1.\] The particular solution is of the form \[Y=At^3+Bt^2+Ct+D.\]

Remember: You need to find \(A\), \(B\), \(C\), \(D\), and the solution to \(y''-4y=0\).