Question

Find the solution to the initial value problem.

(a) 4y'' - y = 0, y(0) = 1, y'(0) = -1
(b) t^2y'' - ty' + 5y = 0, (t > 0) y(1) = 0, y'(1) = 2
"Find the solution to the initial value problem.
(a) 4y'' - y = 0, y(0) = 1, y'(0) = -1
(b) t^2y'' - ty' + 5y = 0, (t > 0) y(1) = 0, y'(1) = 2"

Answer 1

To solve the initial value problems, we use the characteristic equation method for (a) and a similar approach for the Cauchy-Euler equation in (b), then apply initial conditions to find the constants.

Step-by-step explanation:

To solve the initial value problem, we need to follow a few steps. For (a) 4y'' - y = 0, we will assume a solution of the form y = ert, find the characteristic equation, and solve for the roots. For (b) t2y'' - ty' + 5y = 0, we'll use a similar approach and find a solution that fits the given initial conditions.

For (a), the characteristic equation is 4r2-1=0, which gives us r = 1/2 and r = -1/2. Thus, the general solution will be of the form y = C1et/2+C2e-t/2. Applying the initial conditions y(0) = 1 and y'(0) = -1 will allow us to solve for the constants C1 and C2.

For (b), since it is a Cauchy-Euler equation, we assume a solution of the form y = tm, where m is a number we need to determine. By substituting into the equation and comparing coefficients, we can find the values of m and then solve for the specific solution that satisfies the conditions y(1) = 0 and y'(1) = 2.