To solve this problem, we’ll apply Euler's method over two time steps for the given differential equation.

Problem Analysis

The equation given is: $\dot{x} = -t x^2$ with the initial condition: $x(0) = 1$

Euler's method for approximating solutions to differential equations is defined as: $x_{n+1} = x_n + h f(t_n, x_n)$ where $h$ is the step size (here, $h = \Delta t$), and $f(t, x) = -t x^2$ based on the differential equation.

Step-by-Step Solution Using Euler’s Method

1. First Step (at $t = 0$):

   • Initial values: $t_0 = 0$, $x_0 = 1$.
   • Using $f(t_0, x_0) = -t_0 x_0^2 = 0$, we find: $x_1 = x_0 + \Delta t \cdot f(t_0, x_0) = 1 + \Delta t \cdot 0 = 1$
   • So, after the first step: $x(\Delta t) = 1$.

2. Second Step (at $t = \Delta t$):

   • Now, $t_1 = \Delta t$, $x_1 = 1$.
   • Using $f(t_1, x_1) = -t_1 x_1^2 = -\Delta t \cdot 1^2 = -\Delta t$, we find: $x_2 = x_1 + \Delta t \cdot f(t_1, x_1) = 1 + \Delta t \cdot (-\Delta t) = 1 - (\Delta t)^2$

Thus, after two time steps, we have: $x(2\Delta t) = 1 - (\Delta t)^2$

Answer

The correct choice is: $\boxed{x(2\Delta t) = 1 - (\Delta t)^2}$ This matches the third option in the list.

Do you want more details on Euler's method, or have any other questions?

Follow-up Questions

1. How would the solution change if we used a different step size $\Delta t$?
2. What is the general formula for Euler's method, and how does it apply to nonlinear differential equations?
3. How can we improve the accuracy of Euler’s method?
4. What would be the result if we took more steps, say three or four steps?
5. How does Euler's method compare with other methods like the Runge-Kutta methods?

Tip

When using Euler’s method, smaller step sizes (smaller $\Delta t$) generally yield more accurate results, but require more computation steps.