Consider the initial value problem y" + 2y + 5xy = 0, y(0)=-6, y(0)= -7. The first 5 Taylor polynomial approximations of the solution are plotted below. You will compute the terms in these approximations. 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 y),..., in terms of y and y' and enter these in the left column below. Use these formulas to evaluate y (0). y (0), y" (0) (0) for the solution of the IVP given above, and enter these numbers in the middle column. In the rightmost column, enter the corresponding terms of the Taylor series of the solution of the IVP. Remember to use + for all multiplications. Note that an entire row will be marked incorrect if anything in the row is incorrect. { (3) y (0) = -6 -6 y (0)- y" (0) (0)- (0)- ✓ term in Taylor series -6 term in Taylor series term in Taylor series term in Taylor series term in Taylor series

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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**Consider the Initial Value Problem**

\[ y'' + 2y' + 5xy = 0, \quad y(0) = -6, \quad y'(0) = -7. \]

The first 5 Taylor polynomial approximations of the solution are plotted below. You will compute the terms in these approximations.

**Graph Description:**

The graph displays four curves, each representing a Taylor polynomial approximation of different orders (1 term, 2 terms, 3 terms, and 5 terms). The x-axis ranges from approximately -0.6 to 0.6, while the y-axis ranges from -7 to -2. The graph shows how each polynomial approximates the solution of the differential equation around \( x = 0 \).

**Instructions:**

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 the expression, obtain formulas for the derivatives \( y^{(3)}, \ldots, y^{(6)} \), in terms of \( y \) and \( y' \), and enter these in the left column below. Use these formulas to evaluate \( y(0), y'(0), y''(0), \ldots, y^{(4)}(0) \) for the solution of the IVP given above, and enter these numbers in the middle column. In the rightmost column, enter the corresponding terms of the Taylor series of the solution of the IVP.

Remember to use \( * \) for all multiplications.

Note that an entire row will be marked incorrect if anything in the row is incorrect.

**Table for Completion:**

- \( y(0) = \)
  - Value: \(-6\)
  - Term in Taylor series: \(-6\)

- \( y'(0) = \)
  - (To be completed)
  - Term in Taylor series:
  
- \( y'' = \)
  - \( y''(0) = \)
  - Term in Taylor series:

- \( y^{(3)} = \)
  - \( y^{(3)}(0) = \)
  - Term in Taylor series:
  
- \( y^{(4)} = \)
  - \( y^{(4)}(0) = \)
  - Term
Transcribed Image Text:**Consider the Initial Value Problem** \[ y'' + 2y' + 5xy = 0, \quad y(0) = -6, \quad y'(0) = -7. \] The first 5 Taylor polynomial approximations of the solution are plotted below. You will compute the terms in these approximations. **Graph Description:** The graph displays four curves, each representing a Taylor polynomial approximation of different orders (1 term, 2 terms, 3 terms, and 5 terms). The x-axis ranges from approximately -0.6 to 0.6, while the y-axis ranges from -7 to -2. The graph shows how each polynomial approximates the solution of the differential equation around \( x = 0 \). **Instructions:** 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 the expression, obtain formulas for the derivatives \( y^{(3)}, \ldots, y^{(6)} \), in terms of \( y \) and \( y' \), and enter these in the left column below. Use these formulas to evaluate \( y(0), y'(0), y''(0), \ldots, y^{(4)}(0) \) for the solution of the IVP given above, and enter these numbers in the middle column. In the rightmost column, enter the corresponding terms of the Taylor series of the solution of the IVP. Remember to use \( * \) for all multiplications. Note that an entire row will be marked incorrect if anything in the row is incorrect. **Table for Completion:** - \( y(0) = \) - Value: \(-6\) - Term in Taylor series: \(-6\) - \( y'(0) = \) - (To be completed) - Term in Taylor series: - \( y'' = \) - \( y''(0) = \) - Term in Taylor series: - \( y^{(3)} = \) - \( y^{(3)}(0) = \) - Term in Taylor series: - \( y^{(4)} = \) - \( y^{(4)}(0) = \) - Term
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