Find a general solution to the differential equation using the method of variation of parameters. y" +9y = 4 csc 23t The general solution is y(t) =
Find a general solution to the differential equation using the method of variation of parameters. y" +9y = 4 csc 23t The general solution is y(t) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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5Differential equations
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![**Topic: Solving Differential Equations Using the Method of Variation of Parameters**
**Problem:**
Find a general solution to the differential equation using the method of variation of parameters.
\[ y'' + 9y = 4\csc^2(3t) \]
**Solution:**
To find the general solution using the method of variation of parameters, follow these steps:
1. **Homogeneous Solution**:
- First, find the complementary (homogeneous) solution to the homogeneous equation, \( y'' + 9y = 0 \).
- The characteristic equation for this differential equation is:
\[ r^2 + 9 = 0 \]
Solving for \(r\), we get:
\[ r = \pm 3i \]
- This gives the complementary solution:
\[ y_c(t) = c_1 \cos(3t) + c_2 \sin(3t) \]
2. **Particular Solution**:
- Next, find a particular solution \( y_p(t) \) using variation of parameters.
- Assume \( y_p(t) = u_1(t) \cos(3t) + u_2(t) \sin(3t) \), where \( u_1(t) \) and \( u_2(t) \) are functions to be determined.
- Calculate the derivatives \( y_p' \) and \( y_p'' \) and substitute them into the original non-homogeneous equation.
- Use the method of variation of parameters to solve for \( u_1(t) \) and \( u_2(t) \).
3. **General Solution**:
- Once \( u_1(t) \) and \( u_2(t) \) are determined, plug them back into the particular solution.
- The general solution to the differential equation is given by:
\[ y(t) = y_c(t) + y_p(t) \]
**Result:**
Finally, enter your calculated solution into the provided box.
\[ \text{The general solution is } y(t) = \boxed{\quad} \]
*Note: To solve for \( u_1(t) \) and \( u_2(t) \), integration and further algebraic manipulation are required. Specific steps depend on the integrals involved and can be complex.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1d700aa-072a-4ff7-be1d-cdca0f10aa9d%2F78c26b54-2c95-4502-9187-e52c380369f9%2Ffyjky8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Solving Differential Equations Using the Method of Variation of Parameters**
**Problem:**
Find a general solution to the differential equation using the method of variation of parameters.
\[ y'' + 9y = 4\csc^2(3t) \]
**Solution:**
To find the general solution using the method of variation of parameters, follow these steps:
1. **Homogeneous Solution**:
- First, find the complementary (homogeneous) solution to the homogeneous equation, \( y'' + 9y = 0 \).
- The characteristic equation for this differential equation is:
\[ r^2 + 9 = 0 \]
Solving for \(r\), we get:
\[ r = \pm 3i \]
- This gives the complementary solution:
\[ y_c(t) = c_1 \cos(3t) + c_2 \sin(3t) \]
2. **Particular Solution**:
- Next, find a particular solution \( y_p(t) \) using variation of parameters.
- Assume \( y_p(t) = u_1(t) \cos(3t) + u_2(t) \sin(3t) \), where \( u_1(t) \) and \( u_2(t) \) are functions to be determined.
- Calculate the derivatives \( y_p' \) and \( y_p'' \) and substitute them into the original non-homogeneous equation.
- Use the method of variation of parameters to solve for \( u_1(t) \) and \( u_2(t) \).
3. **General Solution**:
- Once \( u_1(t) \) and \( u_2(t) \) are determined, plug them back into the particular solution.
- The general solution to the differential equation is given by:
\[ y(t) = y_c(t) + y_p(t) \]
**Result:**
Finally, enter your calculated solution into the provided box.
\[ \text{The general solution is } y(t) = \boxed{\quad} \]
*Note: To solve for \( u_1(t) \) and \( u_2(t) \), integration and further algebraic manipulation are required. Specific steps depend on the integrals involved and can be complex.*
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