# An Operator Method for Solving Second Order Differential Equations

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} =…

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