| Title | Summary |
|---|---|
| Finding Particular Solutions to Inhomogeneous ODEs | Professor Mattuck (MIT video) shows how to find the general solution of a second-order nonhomogeneous linear equation with constant coefficients when the forcing has characteristic form with degree zero. He uses Key Identity Evaluation (but does not call it that) for the case when the characteristic of the forcing has multiplicity zero. The formulas he gives for the cases when the characteristic of the forcing has multiplicities one and two follow easily from Key Identity Evaluations, but he gives a more complicated derivation of them based on something he calls the "exponential shift formula". |
| Second-Order Linear Equations: Undetermined Coefficients | Paul's On-Line Notes presenting the method of undetermined coefficients for
computing particular solutions to second-order nonhomogeneous linear equations with constant coefficients when the forcing has characteristic form. A recipe for the general form of the particular solution is given, but in a complicated way that is broken into many cases. There is no justification of it. The case when the characteristic of the forcing does not have multiplicity zero is not presented well. |
| Higher-Order Linear Equations: Undetermined Coefficients | Paul's On-Line Notes presenting the method of undetermined coefficients for
computing particular solutions to higher-order nonhomogeneous linear equations with constant coefficients when the forcing has characteristic form. One example is worked out with a forcing that has composite characteristic form, so the problem decouples into three problems --- but this is not explained. |
| Khan Academy: Undertermined Coefficients 1 | Khan Academy video showing how to find the general solution of a second-order nonhomogeneous linear equation with constant coefficients when the characteristic polynomial of the associated homogeneous equation has two real roots and the forcing is an expoential function. A general solution is obtained as the sum of a general solution of the associated homogeneous equation and a particular solution of the nonhomogeneous equation. The particular solution is obtained be "guessing" its form up to an "underterimed coefficient" A and then determining A by plugging this form into the equation. The video does not explain the limitations of this method or how to more generally find the "guess". |
| Kahn Academy: Undertermined Coefficients 2 | Kahn Academy video showing how to find the general solution of a second-order nonhomogeneous linear equation with constant coefficients when the characteristic polynomial of the associated homogeneous equation has two real roots and the forcing is a sine function. The method is the same as in the previous video, but with a more complicated "guess" that has two undetermined coefficients. |
| Kahn Academy: Undertermined Coefficients 3 | Kahn Academy video showing how to find the general solution of a second-order nonhomogeneous linear equation with constant coefficients when the characteristic polynomial of the associated homogeneous equation has two real roots and the forcing is a power function. The method is the same as in the previous video, but with a more complicated "guess" with three undetermined coefficients. |
| Kahn Academy: Undertermined Coefficients 4 | Kahn Academy video showing how to find the general solution of a second-order nonhomogeneous linear equation with constant coefficients when the characteristic polynomial of the associated homogeneous equation has two real roots and the forcing is a the sum of the forcings in the previous three examples. A general solution is obtained as the sum of a general solution of the associated homogeneous equation and the three particular solutions found in the three previous examples. This forcing in an example is what the text calls a composite characteristic forcing. |