To solve the initial value problem $\frac{\mathrm{d}y}{\mathrm{d}x} = x + y$ with $y(0) = -1$ using Picard's method up to the third approximation, follow these steps:

Step 1: Picard's iterative formula

Picard’s method uses the integral form of the differential equation: $y(x) = y(0) + \int_{x_0}^{x} \big(f(t, y(t))\big) \, \mathrm{d}t.$ Here, $f(x, y) = x + y$, and the initial condition is $y(0) = -1$.

Step 2: Iterative process

1. First approximation ($y_0(x)$): Start with the constant function given by the initial condition: $y_0(x) = -1.$

2. Second approximation ($y_1(x)$): Substitute $y_0(x)$ into the integral: $y_1(x) = -1 + \int_0^x \big(t + y_0(t)\big) \, \mathrm{d}t.$ Substituting $y_0(t) = -1$: $y_1(x) = -1 + \int_0^x \big(t - 1\big) \, \mathrm{d}t = -1 + \left[\frac{t^2}{2} - t \right]_0^x.$ Evaluating the integral: $y_1(x) = -1 + \left(\frac{x^2}{2} - x\right) = \frac{x^2}{2} - x - 1.$

3. Third approximation ($y_2(x)$): Substitute $y_1(x)$ into the integral: $y_2(x) = -1 + \int_0^x \big(t + y_1(t)\big) \, \mathrm{d}t.$ Substituting $y_1(t) = \frac{t^2}{2} - t - 1$: $y_2(x) = -1 + \int_0^x \left(t + \frac{t^2}{2} - t - 1\right) \, \mathrm{d}t.$ Simplify the integrand: $t + \frac{t^2}{2} - t - 1 = \frac{t^2}{2} - 1.$ Compute the integral: $y_2(x) = -1 + \int_0^x \left(\frac{t^2}{2} - 1\right) \, \mathrm{d}t = -1 + \left[\frac{t^3}{6} - t\right]_0^x.$ Evaluating: $y_2(x) = -1 + \left(\frac{x^3}{6} - x\right) = \frac{x^3}{6} - x - 1.$

Final Answer

The approximations up to the third step are:

1. $y_0(x) = -1$,
2. $y_1(x) = \frac{x^2}{2} - x - 1$,
3. $y_2(x) = \frac{x^3}{6} - x - 1$.

Would you like a detailed explanation of any step or further assistance?

Here are five related questions to explore:

1. What is the general approach to Picard's method for higher-order approximations?
2. How does Picard's method compare to numerical methods like Euler's?
3. Can we use Picard's method for non-linear differential equations?
4. How does the convergence of Picard's method depend on the initial conditions and the function $f(x, y)$?
5. How is the exact solution related to the Picard approximations?

Tip: When applying Picard's method, always verify that the integral form converges by ensuring $f(x, y)$ is Lipschitz continuous in $y$.