Question

Here is another approach to solving Euler's equation. Transform the equation to a homogeneous, constant coefficient second-order differential equation for Y(t) by setting: x= eᵗ and y(t) = y(ln(x)). Please provide the necessary steps and solution using this approach.

Answer 1

To solve Euler's equation using the given approach, substitute x = e^t and y(t) = y(ln(x)) into the equation to obtain a second-order linear homogeneous differential equation with constant coefficients.

Step-by-step explanation:

To solve Euler's equation using the given approach, we first substitute x = e^t and y(t) = y(ln(x)) into the equation. This gives us:

y''(ln(x))^2 + y'(ln(x)) + y(ln(x)) = 0

Next, we differentiate y(ln(x)) with respect to ln(x) to obtain:

y'(ln(x)) = dy/d(ln(x)) = (dy/dx) / (dx/d(ln(x))) = (dy/dx) / (1/x) = x * dy/dx

Substituting this into the equation, we get:

y''(ln(x))^2 + x * dy/dx + y(ln(x)) = 0

This is a second-order linear homogeneous differential equation with constant coefficients. We can solve it using standard methods.