Final answer:
To solve the given third-order initial value problem with Laplace transforms, the Laplace transform is applied to the differential equation, initial conditions are plugged in, and then the resulting algebraic equation is solved for Y(s). Afterwards, the inverse Laplace transform is used to find y(t), the solution to the original problem.
Step-by-step explanation:
To solve the third-order initial value problem y''' + 3y'' - 10y' - 24y = -72 with the initial conditions y(0) = 3, y'(0) = 4, y''(0) = 46, we start by taking the Laplace transform of each term in the differential equation. Representing the Laplace transform of y(t) as Y(s), we get:
L{y'''} + 3L{y''} - 10L{y'} - 24L{y} = L{-72}
Applying initial conditions and properties of Laplace transforms, we can convert the differential equation into an algebraic equation in terms of Y(s):
s^3Y(s) - s^2y(0) - sy'(0) - y''(0) + 3[s^2Y(s) - sy(0) - y'(0)] - 10[sY(s) - y(0)] - 24Y(s) = -72/s
Substitute the initial conditions given and simplify:
(s^3 + 3s^2 - 10s - 24)Y(s) - (s^2 \* 3 + s \* 4 + 46) + (3s \* 3 + 3 \* 4) - 10 \* 3 = -72/s
This algebraic equation can be solved for Y(s), and then the inverse Laplace transform is applied to find y(t), the solution to the original differential equation.