Question

Solve the initial value problem for the Cauchy-Euler equation:

t²y''(t) + 7ty'(t) + 5y(t) = 0

with initial conditions y(1) = -1 and y'(1) = 13.

Find the solution for y(t).

Answer 1

To solve the Cauchy-Euler equation, we assume a solution of the form y(t) = t^r and find the possible values of r. The general solution is a linear combination of two solutions with different values of r. Substituting the initial conditions gives the specific solution of y(t) = 7t - 8t^5.

Step-by-step explanation:

To solve the Cauchy-Euler equation, we can assume a solution of the form y(t) = t^r. Plugging this into the equation will give us a quadratic equation for r. We solve the equation to find the possible values of r.

Using the initial conditions, we can construct the general solution by taking a linear combination of the two solutions with different values of r. Finally, we substitute the initial conditions to find the specific values of the constants in the general solution.

In this case, the solution for y(t) is y(t) = 7t - 8t^5.