An Operator Method for Solving Second Order Differential Equations, Part 2

In a previous post we discussed an operator method for solving certain second order ordinary differential equations. In this post I’ll explore this operator method a little further.

I first learned about this method from an old book, Higher Mathematics for Engineers and Physicists, by Ivan S. Sokolnikoff and Elizabeth S. Sokolnikoff, McGraw-Hill, 1941. I discovered the book while browsing in a used book store a few years ago, the last time I taught differential equations. You will find some of the ideas behind this post on pages 287 ff.

Consider a first order linear differential equation, with constant coefficients, which is of the form

\$latex y^{prime} + ay = f(x)\$

where a is a constant. A well-known method of solution is to multiply each term by a suitable integrating factor, \$latex e^{ax}\$ in this case, to obtain

\$latex e^{ax}y^{prime} + ae^{ax}y = e^{ax}f(x)\$

Then the left side can…

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