Answer:
u(t) = -(3 + w^2 ) cos t /(1- w^2)cos t + 7 sin t + 8 cos wt /(1- w^2)
Step-by-step explanation:
The characteristic equation is k² + 1 = 0, which leads to k² = -1, resulting in k = ±i.
The roots are k = i or -i.
The general solution takes the form u(x)=C₁cosx+C₂sinx.
Applying the method of undetermined coefficients, we have
Uc(t) = Pcos wt + Qsin wt
Calculating the derivatives gives us Uc’(t) = -Pwsin wt + Qwcos wt
And differentiating again yields Uc’’(t) = -Pw^2cos wt - Qw^2sin wt
With the equation U’’ + u = 8cos wt, we substitute:
-Pw^2cos wt - Qw^2sin wt + Pcos wt + Qsin wt = 8cos wt.
This simplifies to (-Pw^2 + P) cos wt + (-Qw^2 + Q) sin wt = 8cos wt.
From -Pw^2 + P = 8, we find P= 8 /(1- w^2).
From -Qw^2 + Q = 8, we can conclude Q = 0.
Thus, Uc(t) = Pcos wt + Qsin wt = 8 cos wt /(1- w^2).
Combining gives us U(t) = uh(t ) + Uc(t)
= C1cos t + c2 sin t + 8 cos wt /(1- w^2).
Initial conditions yield:
U(0) = C1cos(0) + c2 sin (0) + 8 cos (0) /(1- w^2)
Which leads us to C1 + 8 /(1- w^2) = 5
So C1 = 5 - 8 /(1- w^2) = -(3 + w^2 ) /(1- w^2).
Next, taking the derivative:
U’(t) = -C1 sin t + c2 cos t - 8 w sin wt /(1- w^2).
Evaluating at t = 0 gives us:
U’(0) = -C1 sin (0) + c2 cos (0) - 8 w sin (0) /(1- w^2) = 7.
Thus, c2 = 7.
u(t) = -(3 + w^2 ) cos t /(1- w^2)cos t + 7 sin t + 8 cos wt /(1- w^2)
