### Author Topic: 2020F Test2-ALT-K  (Read 1122 times)

#### RunboZhang

• Sr. Member
•    • Posts: 51
• Karma: 0 ##### 2020F Test2-ALT-K
« on: November 04, 2020, 07:45:49 PM »
$\textbf{Problem:} \\\\ \text{(a) Find the general solution for equation: } \ y'' -4y' + 5y = 30 + 40cos(3t) \\\\ \text{(b) Find solution that satisfies:} \ y(0) = 0, y'(0) = 0$

\textbf{Solution for part (a): } \\\\ \text{Solve for homogenous solution:} \\\\ \begin{gather} \begin{aligned} r^{2} - 4r + 5 = 0 \Longrightarrow r = 2 \pm i \\\\ \end{aligned} \end{gather}

\text{Therefore, } \\\\ \begin{gather} \begin{aligned} y_h = c_1 e^{2t}cos(t) + c_2 e^{2t}sin(t) \end{aligned} \end{gather}

\text{Let } y_p = At+B+Csin(3t)+Dcos(3t) \ \text{ , then we have} \\\\ \begin{gather} \begin{aligned} y'_p &= A+3Ccos(3t) -3Dsin(3t) \\\\ y''_p &= -9Csin(3t)-9Dcos(3t) \end{aligned} \end{gather}

\text{Plug them in the original equation, we have: } \\\\ \begin{gather} \begin{aligned} y'' -4y' + 5y &= -9Csin(3t)-9Dcos(3t) -4(A+3Ccos(3t) -3Dsin(3t)) + 5(At+B+Csin(3t)+Dcos(3t)) \\\\ &= 30 + 40cos(3t) \end{aligned} \end{gather}

\text{Solve for parameter A, B, C, and D, we get } A=0 ,\ B=6,\ C=-3,\ D=-1\\\\ \text{Thus we have: } \\\\ \\\\ \begin{gather} \begin{aligned} Y &= y_h + y_p \\\\ &= c_1 e^{2t}cos(t) + c_2 e^{2t}sin(t) -3sin(3t) -cos(3t) + 6 \end{aligned} \end{gather}

$\textbf{Solution for part (b): } \\\\$

\text{Plug in } y(0)=0 \text{, we get } \ c_1 = -5 \text{, thus we have } \\\\ \begin{gather} \begin{aligned} Y = -5e^{2t}cos(t) + c_2 e^{2t}sin(t) -3sin(3t) -cos(3t) + 6 \end{aligned} \end{gather}

\text{Now consider } y'(0)=0 \text{, then we have:}\\\\ \begin{gather} \begin{aligned} 0 = -5(2e^{2t}cos(t) - e^{2t}sin(t)) + c_2(2e^{2t}sin(t)+e^{2t}cos(t)) - 9cos(3t) + 3sin(3t) \\\\ \Longrightarrow c_2 = 19 \end{aligned} \end{gather}

\text{Thus the solution is:}\\\\ \begin{gather} \begin{aligned} Y = -5e^{2t}cos(t) + 19 e^{2t}sin(t) -3sin(3t) -cos(3t) + 6 \end{aligned} \end{gather}

#### jeffreyz374

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• Karma: 0 ##### Re: 2020F Test2-ALT-K
« Reply #1 on: November 13, 2020, 02:14:36 AM »
I am relieved that our answers to this question agree