Question

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

a. y′′−11y′+36y=0
b. y′′+11y′+36y=0
c. y′′−11y′−36y=0
d. y′′+11y′−36y=0

Answer 1

To transform the given Cauchy-Euler equation to a differential equation with constant coefficients, we can use the substitution x=eᵗ.

Step-by-step explanation:

To transform the given Cauchy-Euler equation to a differential equation with constant coefficients, we can use the substitution x=eᵗ. Let's take the equation y′′−11y′+36y=0 as an example. We first find the derivatives of y with respect to t using the chain rule: y′=dy/dt, y′′=d²y/dt². Then we substitute x=eᵗ and differentiate with respect to t to get dx/dt=eᵗ. We can substitute these expressions for y′ and y′′ into the original equation, replace x with eᵗ, and simplify the equation to get a differential equation with constant coefficients.

Using the substitution x=eᵗ in the given equation y′′−11y′+36y=0, we find that the transformed equation is y′′+11y′−36y=0. Therefore, the correct option is d. y′′+11y′−36y=0.