An Operator Method for Solving Second Order Differential Equations

QED Insight

In talking about power series in a previous post, I mentioned one of their uses: as an aid in solving differential equations. This reminds me of a neat trick for solving some differential equations, which I will discuss in this post.

A standard method for solving linear differential equations with constant coefficients is to assume a trial solution of the form $latex y = e^{rx}$, run it through the differential equation, solve the resulting algebraic equation for r, and then take it from there.

For example, consider the differential equation

$latex y^{prime prime} – y = 0$

Let’s suppose that there is a solution to the differential equation of the form $latex y = e^{rx}$. Then

$latex y^{prime} = re^{rx} quad$ and $latex y^{prime prime} = r^2e^{rx}$

Inserting the expressions for y and $latex y^{prime prime} $ into the differential equation, we get

$latex r^2e^{rx} – e^{rx} =…

View original post 1,088 more words

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