It was stated in the Closure that any given Cauchy-Euler ou equation can be reduced constant-coefficient equation by the change of variables x = e ᵗ In this exercise we ask you to try that idea for some specific cases; in the next exercise we ask for a general proof of the italicized claim. Let y (x(t)) equiv Y(t) and let y' and Y' denote dy/dx and dY/dt, respectively.
(a) Show that the change of variables x = e ᵗ reduces the Cauchy-Euler equation x² yᵖrime prime - x y' - 3y = 0 to the constant- coefficient equation Yᵖrime prime - 2 Y' - 3Y = 0 Thus, show that Y(t) = A e ⁻ᵗ + B e ³ᵗ) Since t = ln(x) show that y(x) = A x ⁻ 1 + B x ³ .