Question

Use the substitution x = eᵗ to transform the given Cauchy-Euler equation to a differential equation with constant coefficients.

(Use yp for dy /dt and ypp for d²y/dt²)

x²y" – 15xy' + 64y = 0

Answer 1

To transform the given Cauchy-Euler equation into a differential equation with constant coefficients, we can use the substitution x = eᵗ. We differentiate x with respect to t and substitute the expressions into the original equation. The resulting equation is e²ᵗy'' - 15eᵗy' + 64y = 0.

Step-by-step explanation:

To transform the Cauchy-Euler equation x²y'' - 15xy' + 64y = 0 into a differential equation with constant coefficients, we can use the substitution x = eᵗ. We need to differentiate x with respect to t to get dx/dt, and then substitute these expressions into the original equation. Let's go through the steps:

1. Differentiate x = eᵗ with respect to t: dx/dt = eᵗ
2. Differentiate dx/dt with respect to t: d²x/dt² = d/dt(eᵗ) = eᵗ
3. Substitute the expressions for x, dx/dt, and d²x/dt² into the original equation: (eᵗ)²y'' - 15(eᵗ)y' + 64y = 0
4. Simplify the equation: e²ᵗy'' - 15eᵗy' + 64y = 0

This resulting equation is now a differential equation with constant coefficients.