3y" +2y + 3xy = 0, y (0) = 4, 4 (0) = –9. The first 5 Taylor polynomial approximations of the solution are plotted below. You will compute the terms in these approximations. 14 1 term 2 terms 12 3 terms 4 terms 5 terms 10 8 -0.4 -0.2 0.0 0.2 0.4 Rewrite the differential equation in the form y" = something, and enter that something in the top slot of the left column. Use y' for y. Then by repeatedly differentiating that expression, obtain formulas for the derivatives y3), ... , column below. Use these formulas to evaluate y (0) , y (0), y" (0), middle column. In the rightmost column, enter the corresponding terms of the Taylor series of the solution of the IVP. y(4), in terms of y and y' and enter these in the left . , y(4) (0) for the solution of the IVP given above, and enter these numbers in the .
3y" +2y + 3xy = 0, y (0) = 4, 4 (0) = –9. The first 5 Taylor polynomial approximations of the solution are plotted below. You will compute the terms in these approximations. 14 1 term 2 terms 12 3 terms 4 terms 5 terms 10 8 -0.4 -0.2 0.0 0.2 0.4 Rewrite the differential equation in the form y" = something, and enter that something in the top slot of the left column. Use y' for y. Then by repeatedly differentiating that expression, obtain formulas for the derivatives y3), ... , column below. Use these formulas to evaluate y (0) , y (0), y" (0), middle column. In the rightmost column, enter the corresponding terms of the Taylor series of the solution of the IVP. y(4), in terms of y and y' and enter these in the left . , y(4) (0) for the solution of the IVP given above, and enter these numbers in the .
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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![**Title: Taylor Polynomial Approximations for Differential Equations**
**Differential Equation:**
\[ y'' + 2y' + 3xy = 0, \quad y(0) = 4, \quad y'(0) = -9. \]
The first 5 Taylor polynomial approximations of the solution are plotted below. You will compute the terms in these approximations.
**Graph Explanation:**
The graph displays five curves, each representing a different Taylor polynomial approximation of the solution to the given differential equation. The curves are color-coded based on the number of terms used in the Taylor series:
- **Blue (1 term)**
- **Green (2 terms)**
- **Red (3 terms)**
- **Cyan (4 terms)**
- **Magenta (5 terms)**
The x-axis ranges from -0.6 to 0.6, and the y-axis ranges from -2 to 14. As more terms are added, the approximation better fits the expected behavior of the solution.
**Instructions for Calculating Terms:**
1. **Rewriting the Differential Equation:**
- Rewrite it in the form \( y'' = \text{something} \), and enter that expression in the left column. Use \( y' \) for derivatives.
2. **Differentiation:**
- By differentiating, obtain forms for \( y^{(3)}, \ldots, y^{(4)} \) in terms of \( y \) and \( y' \), then enter these in the left column.
3. **Evaluation:**
- Evaluate \( y(0), y'(0), y''(0), \ldots, y^{(4)}(0) \) for the initial value problem (IVP) and enter these numbers in the middle column.
4. **Taylor Series Terms:**
- Enter corresponding terms of the Taylor series on the right.
**Note:**
- Use \(*\) for all multiplications.
- Accuracy is crucial; the incorrectness in any row results in a marked incorrect entry.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb493ecdd-cbdc-400d-a05a-de2010eb2d52%2Ff11a8b7c-211c-475b-8bc5-341f935f6826%2F6dtj1f_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Taylor Polynomial Approximations for Differential Equations**
**Differential Equation:**
\[ y'' + 2y' + 3xy = 0, \quad y(0) = 4, \quad y'(0) = -9. \]
The first 5 Taylor polynomial approximations of the solution are plotted below. You will compute the terms in these approximations.
**Graph Explanation:**
The graph displays five curves, each representing a different Taylor polynomial approximation of the solution to the given differential equation. The curves are color-coded based on the number of terms used in the Taylor series:
- **Blue (1 term)**
- **Green (2 terms)**
- **Red (3 terms)**
- **Cyan (4 terms)**
- **Magenta (5 terms)**
The x-axis ranges from -0.6 to 0.6, and the y-axis ranges from -2 to 14. As more terms are added, the approximation better fits the expected behavior of the solution.
**Instructions for Calculating Terms:**
1. **Rewriting the Differential Equation:**
- Rewrite it in the form \( y'' = \text{something} \), and enter that expression in the left column. Use \( y' \) for derivatives.
2. **Differentiation:**
- By differentiating, obtain forms for \( y^{(3)}, \ldots, y^{(4)} \) in terms of \( y \) and \( y' \), then enter these in the left column.
3. **Evaluation:**
- Evaluate \( y(0), y'(0), y''(0), \ldots, y^{(4)}(0) \) for the initial value problem (IVP) and enter these numbers in the middle column.
4. **Taylor Series Terms:**
- Enter corresponding terms of the Taylor series on the right.
**Note:**
- Use \(*\) for all multiplications.
- Accuracy is crucial; the incorrectness in any row results in a marked incorrect entry.
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