Author Topic: TUT0801 QUIZ5  (Read 539 times)

taojinwe

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TUT0801 QUIZ5
« on: November 01, 2019, 01:17:17 PM »
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⎮