Homogeneous Linear Equations

Roots of the characteristic polynomial ar^2 + br + c = 0 solve ax'' + bx' + cx = 0, giving the damped and undamped motion of a mass on a spring.

The ODE Project, Homogeneous Linear Equations

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Exercise 38

An important class of second-order linear differential equations are equations of the form where $t \gt 0$ and $\alpha$ and $\beta$ are real constants. An equation of this form is called an Euler equation.

Show that the substitution $x = \ln t$ transforms an Euler equation into an equation of constant coefficients.
Solve the equation

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