show that the inhomogeneous equation y"+y=secx, has the homogeneous solutions y cos x and y2 sinx. (Plug them into the homogeneous equation and show they check.) Find the Wronskian for o conditions are given. formally what constants you use are arbitrary, but for grading purposes use A as the coefficient for cosine and B as the coefficient for sine in your general solution. The Wronskian Wicos x, sin x) 1 =(x)-log(sec(x)) x The general solution is (x)= −cos(x(log(sec(x))))+xsin(x) +4-cos(x) + B-sin(x)
show that the inhomogeneous equation y"+y=secx, has the homogeneous solutions y cos x and y2 sinx. (Plug them into the homogeneous equation and show they check.) Find the Wronskian for o conditions are given. formally what constants you use are arbitrary, but for grading purposes use A as the coefficient for cosine and B as the coefficient for sine in your general solution. The Wronskian Wicos x, sin x) 1 =(x)-log(sec(x)) x The general solution is (x)= −cos(x(log(sec(x))))+xsin(x) +4-cos(x) + B-sin(x)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Problem Statement:**
Show that the inhomogeneous equation \( y'' + y = \sec x \), has the homogeneous solutions \( y_1 = \cos x \) and \( y_2 = \sin x \). (Plug them into the homogeneous equation and show they check.) Find the Wronskian for this pair of solutions, determine the functions \( u(x) \) and \( v(x) \) to apply the variation of parameters method, and the general solution. No conditions are given.
Normally what constants you use are arbitrary, but for grading purposes use \( A \) as the coefficient for cosine and \( B \) as the coefficient for sine in your general solution.
**Solution Steps:**
1. **Wronskian Calculation:**
\[
W(\cos x, \sin x) = 1 \quad \text{(Checkmark indicating correct response)}
\]
2. **Determine Functions:**
- \( u(x) = \log(\sec(x)) \) (Incorrect answer indicated)
- \( v(x) = x \) (Correct answer indicated)
3. **General Solution:**
\[
y = \left(-\cos \left(x \left(\log(\sec(x))\right) \right) \right) + x\sin(x) + A\cdot \cos(x) + B\cdot \sin(x) \quad \text{(Incorrect answer indicated)}
\]
**Notes on Solution:**
The problem involves applying the method of variation of parameters to solve a differential equation. It requires finding particular solutions using given homogeneous solutions \( \cos x \) and \( \sin x \) and calculating the Wronskian determinant to find functions \( u(x) \) and \( v(x) \).
The calculation of the Wronskian was done correctly, resulting in the value of 1. However, errors are indicated in determining \( u(x) \) and in the final general solution.
It is crucial to verify each step, especially when applying the variation of parameters method, to ensure accuracy in calculating the functions and the final solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8e0ad34-47bd-4edd-8e64-7fbb4be3cef6%2Fc26b3650-4b09-497c-93ce-94bd18768a58%2Fgx43ij8_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Show that the inhomogeneous equation \( y'' + y = \sec x \), has the homogeneous solutions \( y_1 = \cos x \) and \( y_2 = \sin x \). (Plug them into the homogeneous equation and show they check.) Find the Wronskian for this pair of solutions, determine the functions \( u(x) \) and \( v(x) \) to apply the variation of parameters method, and the general solution. No conditions are given.
Normally what constants you use are arbitrary, but for grading purposes use \( A \) as the coefficient for cosine and \( B \) as the coefficient for sine in your general solution.
**Solution Steps:**
1. **Wronskian Calculation:**
\[
W(\cos x, \sin x) = 1 \quad \text{(Checkmark indicating correct response)}
\]
2. **Determine Functions:**
- \( u(x) = \log(\sec(x)) \) (Incorrect answer indicated)
- \( v(x) = x \) (Correct answer indicated)
3. **General Solution:**
\[
y = \left(-\cos \left(x \left(\log(\sec(x))\right) \right) \right) + x\sin(x) + A\cdot \cos(x) + B\cdot \sin(x) \quad \text{(Incorrect answer indicated)}
\]
**Notes on Solution:**
The problem involves applying the method of variation of parameters to solve a differential equation. It requires finding particular solutions using given homogeneous solutions \( \cos x \) and \( \sin x \) and calculating the Wronskian determinant to find functions \( u(x) \) and \( v(x) \).
The calculation of the Wronskian was done correctly, resulting in the value of 1. However, errors are indicated in determining \( u(x) \) and in the final general solution.
It is crucial to verify each step, especially when applying the variation of parameters method, to ensure accuracy in calculating the functions and the final solution.
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