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

QED Insight

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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