Question

Find the solution of the initial value problem y''' - 8y'' + 4y' - 32y = sec^2 t with y(0) = 2, y'(0) = 7, y''(0) = 94.

a) y(t) = sin t + 2 cos t
b) y(t) = 2 cos t + sin t
c) y(t) = cos t + 2 sin t
d) y(t) = 2 sin t + cos t

Answer 1

To find the solution of the initial value problem, we use the method of undetermined coefficients. The particular solution is found by assuming a form and substituting it into the differential equation. .The correct answer is d) y(t) = 2sin(t) + cos(t).

Step-by-step explanation:

To solve the given initial value problem, we can use the method of undetermined coefficients. First, we find the characteristic equation by substituting y = e^(rt) into the differential equation: r^3 - 8r^2 + 4r - 32 = 0. By solving this equation, we find that the roots are r = 4, -2 ± 4i.

Next, we assume a particular solution of the form y(t) = Asec^2(t). We take the derivatives of y(t) and substitute them into the differential equation to find the value of A.

After finding the value of A, we combine the particular solution and the homogeneous solutions to obtain the general solution.The specific solution is obtained by combining the particular solution with the homogeneous solutions and applying the initial conditions

Finally, we use the initial conditions y(0) = 2, y'(0) = 7, and y''(0) = 94 to find the specific solution of the initial value problem. The correct answer is d) y(t) = 2sin(t) + cos(t).