To solve the first-order differential equation $\frac{dy}{dx} = x^2 + y$ using Euler's method with initial conditions $y(0) = 1$, step sizes $h = 0.01$ and $h = 0.05$, and $x(0) = 0$, we proceed as follows:

Problem Setup:

The differential equation is:

$\frac{dy}{dx} = x^2 + y$

With the initial condition $y(0) = 1$, we need to apply Euler's method for two different step sizes $h = 0.01$ and $h = 0.05$.

Euler's Method:

Euler's method is a numerical technique for solving differential equations. Given the differential equation $\frac{dy}{dx} = f(x, y)$ and initial values $x_0$ and $y_0$, the formula for the next point is:

$y_{n+1} = y_n + h \cdot f(x_n, y_n)$ $x_{n+1} = x_n + h$

Where $f(x_n, y_n) = x_n^2 + y_n$.

Step-by-Step Approach:

1. Start at $x = 0$ with $y(0) = 1$.
2. Apply the Euler's formula to compute the next values of $y$ and $x$ for each step.

Now, I will compute the values for each step size. Let's begin with $h = 0.01$ and $h = 0.05$. I'll calculate the values numerically for the first few steps.

For $h = 0.01$:

1. At $x_0 = 0$, $y_0 = 1$: $y_1 = 1 + 0.01 \cdot (0^2 + 1) = 1.01$ $x_1 = 0 + 0.01 = 0.01$
2. At $x_1 = 0.01$, $y_1 = 1.01$: $y_2 = 1.01 + 0.01 \cdot (0.01^2 + 1.01) = 1.01 + 0.01 \cdot (0.0001 + 1.01) = 1.010 + 0.0101 = 1.0201$ $x_2 = 0.01 + 0.01 = 0.02$

For $h = 0.05$:

1. At $x_0 = 0$, $y_0 = 1$: $y_1 = 1 + 0.05 \cdot (0^2 + 1) = 1.05$ $x_1 = 0 + 0.05 = 0.05$
2. At $x_1 = 0.05$, $y_1 = 1.05$: $y_2 = 1.05 + 0.05 \cdot (0.05^2 + 1.05) = 1.05 + 0.05 \cdot (0.0025 + 1.05) = 1.05 + 0.05 \cdot 1.0525 = 1.05 + 0.052625 = 1.102625$ $x_2 = 0.05 + 0.05 = 0.10$

I can continue calculating more steps or generate tables for these computations. Would you like to see a complete table for both step sizes $h = 0.01$ and $h = 0.05$ up to a specific value of $x$? Or would you like further details on any part?

Follow-Up Questions:

1. Would you like to see the results for a larger interval of $x$?
2. Should we use a different numerical method like Runge-Kutta for comparison?
3. Do you need help visualizing these results on a graph?
4. Should we investigate the effect of using smaller or larger step sizes?
5. Would you like an explanation of how Euler’s method accuracy is affected by $h$?

Tip: Euler's method is easy to implement but can be inaccurate for stiff equations or when larger step sizes are used. Smaller step sizes yield more accurate results.