Question

Given a Cauchy-Euler equation x²y'' - 11xy' - 36y = 0, use the substitution x = et to transform it into a differential equation with constant coefficients.

Answer 1

To transform the given Cauchy-Euler equation x²y'' - 11xy' - 36y = 0 into a differential equation with constant coefficients, we can use the substitution x = et.

Step-by-step explanation:

To transform the given Cauchy-Euler equation x²y'' - 11xy' - 36y = 0 into a differential equation with constant coefficients, we can use the substitution x = et.

Let's start by finding the derivatives of y with respect to x using the chain rule:

y' = dy/dx = (dy/dt) / (dx/dt) = (dy/dt) / (e)

y'' = d²y/dx² = (dy'/dt) / (dx/dt) = (d/dt)(dy') / (dx/dt) = (d/dt)((dy/dt) / (e)) / (e)

Now, substitute these derivatives into the original equation and simplify to obtain a differential equation with constant coefficients.