Question

Solve the given initial-value problem. y',' + 18y" + 81y' = 0, y(0) = 0, y'(0) = 1, y''(0) =-6

Answer 1

To solve the given initial-value problem, we can use the characteristic equation method. The solution to the initial-value problem is y(t) = c₁e^(-9t) + c₂te^(-9t).

Step-by-step explanation:

To solve the given initial-value problem y'', + 18y' + 81y' = 0, y(0) = 0, y'(0) = 1, y''(0) =-6, we can use the characteristic equation method.

First, we assume that the solution is of the form y(t) = e^(rt).

Plugging this into the differential equation, we get the characteristic equation r^2 + 18r + 81 = 0.

Factoring the equation, we have (r + 9)^2 = 0.

So r = -9.

Therefore, the solution to the initial-value problem is y(t) = c₁e^(-9t) + c₂te^(-9t).