An Operator Method For Solving Second Order Differential Equations, Part 3: Wild Speculation

QED Insight

In the twoprevious posts in this series we explored a method for solving second order linear differential equations with constant coefficients that is different from the standard textbook methods taught nowadays. I found the method in a 1941 book (or see here) by the Sokolnikoffs.

The key point of the method, as we learned, is the identification of the action of the inverse of the differential operator 1/(D + a)

$latex dfrac{1}{D + a} , f(x)$

with the action of the integral operator

$latex e^{-ax} int e^{ax} f(x) , {rm d} x$

The previous two posts described solid, well-established mathematics. But now let’s go out on a limb.

Time for some wild speculation

When I see the form of the operator

$latex dfrac{1}{D + a}$

which can also be written as

$latex dfrac{1}{a} cdot dfrac{1}{1 – (-D/a)}$

I can’t help but think of the formula for…

View original post 797 more words


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