dy/dx=x^2+y,y(0)=1,h=0.01,0.05 using euler theorem

To solve the differential equation \( \frac{dy}{dx} = x^2 + y \) using Euler's method, we proceed step-by-step. The initial condition is \( y(0) = 1 \), and we need to find \( y \) for \( h = 0.01 \) and \( h = 0.05 \).

Euler's method approximates the solution to a differential equation by using the formula:
\[
y_{n+1} = y_n + h \cdot f(x_n, y_n)
\]
where \( f(x, y) = x^2 + y \) in this case, \( h \) is the step size, and \( y_n \) is the approximation of \( y \) at step \( n \).

### Step 1: For \( h = 0.01 \)

Given \( h = 0.01 \), let's compute the first few steps.

1. **At \( x_0 = 0 \), \( y_0 = 1 \)** (initial condition):
   \[
   y_1 = y_0 + h \cdot f(x_0, y_0) = 1 + 0.01 \cdot (0^2 + 1) = 1 + 0.01 \cdot 1 = 1.01
   \]

2. **At \( x_1 = 0.01 \), \( y_1 = 1.01 \)**:
   \[
   y_2 = y_1 + h \cdot f(x_1, y_1) = 1.01 + 0.01 \cdot (0.01^2 + 1.01) = 1.01 + 0.01 \cdot (0.0001 + 1.01) = 1.01 + 0.01 \cdot 1.0101 = 1.020101
   \]

### Step 2: For \( h = 0.05 \)

Now with \( h = 0.05 \), let's compute the first step.

1. **At \( x_0 = 0 \), \( y_0 = 1 \)** (initial condition):
   \[
   y_1 = y_0 + h \cdot f(x_0, y_0) = 1 + 0.05 \cdot (0^2 + 1) = 1 + 0.05 \cdot 1 = 1.05
   \]

2. **At \( x_1 = 0.05 \), \( y_1 = 1.05 \)**:
   \[
   y_2 = y_1 + h \cdot f(x_1, y_1) = 1.05 + 0.05 \cdot (0.05^2 + 1.05) = 1.05 + 0.05 \cdot (0.0025 + 1.05) = 1.05 + 0.05 \cdot 1.0525 = 1.102625
   \]

### Summary of Results:
- For \( h = 0.01 \), the first two values are:
  - \( y_1 = 1.01 \)
  - \( y_2 = 1.020101 \)
  
- For \( h = 0.05 \), the first two values are:
  - \( y_1 = 1.05 \)
  - \( y_2 = 1.102625 \)

Would you like me to go further or provide more details?

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Here are 5 related questions:
1. What is the general formula for Euler's method in solving differential equations?
2. How does the accuracy of Euler's method change with step size \( h \)?
3. Can you explain how to apply Euler's method to a second-order differential equation?
4. What is the relationship between Euler's method and Runge-Kutta methods?
5. How can we improve the accuracy of the solution using a smaller step size?

**Tip:** A smaller step size \( h \) in Euler's method generally leads to more accurate results but increases the computational effort.

dy/dx = x^2 + y, y(0) = 1, h = 0.01, 0.05 using Euler theorem

Solve the differential equation dy/dx = x^2 + y with initial condition y(0) = 1 using Euler's method. This step-by-step solution includes results for step sizes h = 0.01 and h = 0.05, demonstrating how Euler's method approximates solutions to differential equations. Suitable for undergraduate students studying calculus or numerical methods.

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