Final answer:
To solve the initial value problem y⁴ - 16y = 0 with given initial conditions, we can use the Laplace transform. By taking the Laplace transform of the differential equation and using the initial conditions, we can solve for the Laplace transform of y(t) and then take the inverse transform to find the solution y(t). The solution is y(t) = 37e^(4t) + 26 + 68cos(2t) + 8sin(2t) + 20t sin(2t).
Step-by-step explanation:
We are given the initial value problem y⁴ - 16y = 0, with initial conditions y(0) = 37, y'(0) = 26, y''(0) = 68, and y'''(0) = 40. To solve this using the Laplace transform, we will first take the Laplace transform of the given differential equation and use the initial conditions to find the transformed equation. Then, we will solve the resulting equation for Y(s), the Laplace transform of y(t). Finally, we will take the inverse Laplace transform to find the solution y(t).
Taking the Laplace transform of y⁴ - 16y = 0, we get s⁴Y(s) - 16sY(s) - 37 + 26s + 68s² + 40s³ = 0. Solving this equation for Y(s), we find Y(s) = (37 - 26s - 68s² - 40s³) / (s⁴ - 16s). Taking the inverse Laplace transform, we obtain the solution y(t) = Laplace inverse {(37 - 26s - 68s² - 40s³) / (s⁴ - 16s)}.
Using partial fraction decomposition, we can express Y(s) as Y(s) = 37 / (s - 4) + 26 / s + 68s / (s² + 4) + 40s² / (s² + 4)². Taking the inverse Laplace transform of each term, we get y(t) = 37e^(4t) + 26 + 68cos(2t) + 8sin(2t) + 20t sin(2t).