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,⇒k²=−1,⇒ k=±i.
the roots are k = i or -i
the solution has the form u(x)=C₁cosx+C₂sinx.
Using undetermined coefficient method
Uc(t) = Pcos wt + Qsin wt
Uc’(t) = -Pwsin wt + Qwcos wt
Uc’’(t) = -Pw^2cos wt - Qw^2sin wt
U’’ + u = 8cos wt
-Pw^2cos wt - Qw^2sin wt + Pcos wt + Qsin wt = 8cos wt
(-Pw^2 + P) cos wt + (- Qw^2 + Q ) sin wt = 8cos wt
-Pw^2 + P = 8 which implies P= 8 /(1- w^2)
- Qw^2 + Q = 8 which implies Q = 0
Uc(t) = Pcos wt + Qsin wt = 8 cos wt /(1- w^2)
U(t) = uh(t ) + Uc(t)
= C1cos t + c2 sin t + 8 cos wt /(1- w^2)
Initial value problem
U(0) = C1cos(0) + c2 sin (0) + 8 cos (0) /(1- w^2)
C1 + 8 /(1- w^2) = 5
C1 = 5 -8 /(1- w^2) = -(3 + w^2 ) /(1- w^2)
U’(t) = -C1 sin t + c2 cos t - 8 w sin wt /(1- w^2)
U’(0) = -C1 sin (0) + c2 cos (0) - 8 w sin (0) /(1- w^2) = 7
c2 = 7
u(t) = -(3 + w^2 ) cos t /(1- w^2)cos t + 7 sin t + 8 cos wt /(1- w^2)
