Undetermined Coefficients: Solving non-homogeneous ODEs

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How can we solve an ordinary differential equation (ODE) like y''-2y'-3y=3e^2t. The problem is the non-homogeneity on the right hand side. If this was homogeneous and just zero, then previously in our playlist we've studied how to solve constant coefficient homogeneous equations. To deal with the non-homogeneity we use a method called undetermined coefficients. The big idea is to guess a so called particular solution with the same basic form as the right hand side, but with undetermined coefficients. Substitute everything in, and you can solve for the constants in your particular solution. Finally, the general solution is the sum of the homogeneous and the particular solution. I show all the standard guesses and how to deal with situations where there are repeats to the homogeneous in the particular.

0:00 Non-homogeneous ODEs
0:50 Particular vs Homogeneous Solutions
4:00 Finding the Particular Solution
6:45 Second Example
9:15 Chart of standard guesses
11:00 Third Example

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