TUT0801 QUIZ5

[taojinwe]

Question: Find the solution of  y''+9y=9sec²(3t) 0<t< 𝞹/6

We first find the homogenous solution of y''+9y=0

r²+9=0

Then r₁=3i and r₂=-3i

Yc(t)=C₁y₁(t)+C₂y₂(t)

=C₁cos3t+C₂sin3t

W=y1 × y2' - y2 × y1' = cos3t×cos3t-sin3t×(-sin3t)=3

Let Yp(t)= 𝝁1y1+ 𝝁2y2

𝝁1=-∫ (sin3t × 9sec²(3t))/3 dt

=- ∫3sin3t × [1/cos²(3t)] dt

=-3 ∫(sec3t × tan3t)dt

= -sec3t

Therefore 𝝁1 = -sec3t

𝝁2 =∫(cos3t × 9sec²(3t))/3 dt

=∫3 cost3t × (1/cos²(3t)dt

= ln⎮sec3t+tant3t⎮

Therefore 𝝁2 = ln⎮sec3t+tan3t⎮

Yp(t)= 𝝁1y1+ 𝝁2y2

=cos3t(-sec3t)+sin3t × ln⎮sec3t+tan3t⎮

Y(t)=Yc(t)+Yp(t)

=C₁cos3t+C₂sin3t+cos3t(-sec3t)+sin3t ln⎮sec3t+tan3t⎮