Question

Use Euler's method to approximate the solutions for each of the following initial-value problem y ′=te 3t −2y,y(0)=0, for 0≤t≤1 with h=0.5.

Answer 1

To approximate the solutions for the given initial-value problem using Euler's method, follow these steps.

Step-by-step explanation:

To approximate the solutions for the given initial-value problem, we can use Euler's method. The equation is y' = te^(3t) - 2y, with the initial condition y(0) = 0 and a step size of h = 0.5. Using Euler's method, we can approximate the values of y at different time points. Here are the steps to do so:

1. Start with the given initial condition y(0) = 0.
2. For each time point, calculate the slope using the derivative equation: f(t, y) = te^(3t) - 2y.
3. Multiply the slope by the step size h = 0.5 to get the change in y: dy = f(t, y) * h.
4. Update the value of y using the formula: y_new = y_old + dy.
5. Repeat steps 2-4 for each time point until the desired range is covered.