3. y' = y³, y(0) = 1

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Needed some help and clarifications on #3 and #5 if possible. Thank You!

### Finding Solutions

For each of the initial value problems:

\[ y' = f(t, y), \]
\[ y(t_0) = y_0 \]

in [Exercise Group 1.4.6.1–6](#), follow these steps:

1. **Write Euler's Method Iteration:**  
   Write the Euler's method iteration \( Y_{k+1} = Y_k + h f(t_k, Y_k) \) for the given problem, identifying the values \( t_0 \) and \( y_0 \).

2. **Compute Approximations:**  
   Using a step size of \( h = 0.1 \), compute the approximations for \( Y_1, Y_2, \) and \( Y_3 \).

3. **Solve Analytically:**  
   Solve the problem analytically if possible. If it is not possible for you to find the analytic solution, use Sage.

4. **Construct Error Table:**  
   Use the results from steps (c) and (d) to construct a table of errors for \( Y_i - y_i \) for \( i = 1, 2, 3 \).

**Problems:**

1. \( y' = -2y, \quad y(0) = 0 \)
2. \( y' = ty, \quad y(0) = 1 \)
3. \( y' = y^3, \quad y(0) = 1 \)
4. \( y' = y, \quad y(0) = 1 \)
5. \( y' = y + t, \quad y(0) = 2 \)
6. \( y' = 1/y, \quad y(0) = 2 \)

For additional insight, refer to the provided Hint, which may guide your understanding and computation.

---

**Note:** Be sure to implement precise calculations and check each step for accuracy to ensure that educational understanding is maximized. Euler's method is a powerful tool for numerically solving differential equations and understanding its application significantly aids in comprehending more advanced mathematical concepts.
Transcribed Image Text:### Finding Solutions For each of the initial value problems: \[ y' = f(t, y), \] \[ y(t_0) = y_0 \] in [Exercise Group 1.4.6.1–6](#), follow these steps: 1. **Write Euler's Method Iteration:** Write the Euler's method iteration \( Y_{k+1} = Y_k + h f(t_k, Y_k) \) for the given problem, identifying the values \( t_0 \) and \( y_0 \). 2. **Compute Approximations:** Using a step size of \( h = 0.1 \), compute the approximations for \( Y_1, Y_2, \) and \( Y_3 \). 3. **Solve Analytically:** Solve the problem analytically if possible. If it is not possible for you to find the analytic solution, use Sage. 4. **Construct Error Table:** Use the results from steps (c) and (d) to construct a table of errors for \( Y_i - y_i \) for \( i = 1, 2, 3 \). **Problems:** 1. \( y' = -2y, \quad y(0) = 0 \) 2. \( y' = ty, \quad y(0) = 1 \) 3. \( y' = y^3, \quad y(0) = 1 \) 4. \( y' = y, \quad y(0) = 1 \) 5. \( y' = y + t, \quad y(0) = 2 \) 6. \( y' = 1/y, \quad y(0) = 2 \) For additional insight, refer to the provided Hint, which may guide your understanding and computation. --- **Note:** Be sure to implement precise calculations and check each step for accuracy to ensure that educational understanding is maximized. Euler's method is a powerful tool for numerically solving differential equations and understanding its application significantly aids in comprehending more advanced mathematical concepts.
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