Question

Use the substitution x = et to transform the given Cauchy-Euler equation to a differential equation with constant coefficients (using yp for dy/dt and ypp for d²y/dt²).

a. x²ypp - 11xy' - 36y = 0
b. x²y'' - 11xy' - 36y = 0
c. x²ypp + 11xy' + 36y = 0
d. x²y'' + 11xy' + 36y = 0

Answer 1

To transform the Cauchy-Euler equation to a differential equation with constant coefficients, substitute x = e^t.

Step-by-step explanation:

To transform the given Cauchy-Euler equation to a differential equation with constant coefficients, we can use the substitution x = e^t. The Cauchy-Euler equation is of the form x^2y'' - 11xy' - 36y = 0. Using the substitution, we have (e^t)^2y'' - 11(e^t)y' - 36y = 0. Simplifying this, we get e^2t y'' - 11e^t y' - 36y = 0, which is a differential equation with constant coefficients.