5. Solve the given equation by using the variation of parameter. y" + y = sec(x) tan (x). Using (i) Complex ミ○ -> yc=Ci Cos . + Ca Sin ... (ii) + Uz. ya Variation of Parameter w=1 第| u, 11以 yı ya U, If yå y2 y: f コ ly,' yo Solve
5. Solve the given equation by using the variation of parameter. y" + y = sec(x) tan (x). Using (i) Complex ミ○ -> yc=Ci Cos . + Ca Sin ... (ii) + Uz. ya Variation of Parameter w=1 第| u, 11以 yı ya U, If yå y2 y: f コ ly,' yo Solve
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Differential Equations: Please show all work so I can practice/understand, thank you =]
![**Title: Solving Differential Equations Using Variation of Parameters**
**Problem Statement:**
Solve the given equation using the variation of parameters:
\[ y'' + y = \sec(x) \tan(x) \]
**Solution Steps:**
**Step 1: Homogeneous Solution (yc)**
\[ y_c = y_h + y_g = 0 \]
\[ y_c = C_1 \cos x + C_2 \sin x \]
**Step 2: Particular Solution (yp) Using Variation of Parameters**
\[ y_p = U_1 y_1 + U_2 y_2 \]
Here,
\[ W = \begin{vmatrix}
y_1 & y_2 \\
y_1' & y_2'
\end{vmatrix} \]
where \( y_1 \) and \( y_2 \) are solutions to the homogeneous equation.
**Variation of Parameters:**
For \( U_1 \):
\[ U_1' = \frac{\begin{vmatrix}
0 & y_2 \\
f & y_2'
\end{vmatrix}}{W} \]
For \( U_2 \):
\[ U_2' = \frac{\begin{vmatrix}
y_1 & 0 \\
y_1' & f
\end{vmatrix}}{W} \]
Here, \( f = \sec(x) \tan(x) \) and \( W \) is the Wronskian determinant of the solutions to the homogeneous equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c88e3a7-d6d9-4180-9e78-6a7df1a2f887%2Faf8f4a1a-eba3-4b45-b8a3-2c4cad0ba0dc%2Fd4wecld_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving Differential Equations Using Variation of Parameters**
**Problem Statement:**
Solve the given equation using the variation of parameters:
\[ y'' + y = \sec(x) \tan(x) \]
**Solution Steps:**
**Step 1: Homogeneous Solution (yc)**
\[ y_c = y_h + y_g = 0 \]
\[ y_c = C_1 \cos x + C_2 \sin x \]
**Step 2: Particular Solution (yp) Using Variation of Parameters**
\[ y_p = U_1 y_1 + U_2 y_2 \]
Here,
\[ W = \begin{vmatrix}
y_1 & y_2 \\
y_1' & y_2'
\end{vmatrix} \]
where \( y_1 \) and \( y_2 \) are solutions to the homogeneous equation.
**Variation of Parameters:**
For \( U_1 \):
\[ U_1' = \frac{\begin{vmatrix}
0 & y_2 \\
f & y_2'
\end{vmatrix}}{W} \]
For \( U_2 \):
\[ U_2' = \frac{\begin{vmatrix}
y_1 & 0 \\
y_1' & f
\end{vmatrix}}{W} \]
Here, \( f = \sec(x) \tan(x) \) and \( W \) is the Wronskian determinant of the solutions to the homogeneous equation.
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