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.