Toronto Math Forum
MAT244-2018S => MAT244--Tests => Term Test 1 => Topic started by: Victor Ivrii on February 15, 2018, 05:12:34 PM
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Find the general solution for equation
\begin{equation*}
y''(t)+8y'(t)+25y(t)=9 e^{-4t}+ 104\sin(3t).
\end{equation*}
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*I will type up the solutions soon*
(https://scontent-yyz1-1.xx.fbcdn.net/v/t34.0-12/28053566_10213726400345645_1400938461_n.png?oh=6d09f74d06187a4510a0d7675798787f&oe=5A87762C)
*Typed solutions to come soon*
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Characteristic equation:
$r^2+8r+25=0$
r = $-4 +3i, -4-3i$ (using quadratic equation)
Homogeneous solution:
$y_c(t) = c_1e^{-4t}cos(3t) + c_2e^{-4t}sin(3t)$
Particular solutions:
First Particular:
$Y = Ae^{-4t}$
$Y' = -4Ae^{-4t}$
$Y'' = 16Ae^{-4t}$
$A(16+8(-4)+25)=9$
$9A=9$
$A=1$
Second Particular:
$Y = Asin(3t)+Bcos(3t)$
$Y' = 3Acos(3t)-3Bsin(3t)$
$Y'' = -9Asin(3t)-9Bcos(3t)$
Plug into the given equation:
sines:
$-9A+8(-3B)+25A = 104$
$16A-24B=104$
$2A-3B=13$
cosines:
$-9B+8(3A)+25B = 0$
$16B+24A=0$
$2B+3A=0$
==> $A=2, B=-3$
General solution:
$y(t) = c_1e^{-4t}cos(3t) + c_2e^{-4t}sin(3t) + e^{-4t} + 2sin(3t) -3cos(3t)$