Use the initial condition y(0)=4 and step size h=.25, calculate the backward Euler on [0,1] for the ODE y'=10y-10y^2. Solve the ODE exactly and compute the errors at t=1.
To use the backward Euler method to solve the given ODE, we need to use the recursive formula:
y_(n+1) = y_n + h * f(t_(n+1), y_(n+1))
where:
- y_n is the numerical approximation of y at time t_n.
- y_(n+1) is the numerical approximation of y at time t_(n+1) = t_n + h = t_n + 0.25.
- h is the step size, which is 0.25 in this case.
- f(t,y) is the derivative of y with respect to t, which is given by f(t,y) = 10y - 10y^2.
We start with the initial condition y(0) = 4, and we want to calculate the value of y at t = 1 using the backward Euler method.
Let's calculate the value of y at each time step using the backward Euler method:
At t = 0:
- y_0 = 4 (given initial condition).
At t = 0.25:
- Compute f(t_1, y_1) = 10 * y_1 - 10 * y_1^2.
- From the recursive formula, we have: y_1 = y_0 + h * f(t_1, y_1).
- Substitute the values: y_1 = 4 + 0.25 * (10 * y_1 - 10 * y_1^2).
- This equation needs to be solved to find y_1.
Similarly, we can calculate the next values of y at each time step using the same process.
To calculate the exact solution of the ODE, we need to solve the differential equation analytically. In this case, the ODE y' = 10y - 10y^2 is separable. By separating the variables and integrating, we can obtain the exact solution y(t).
Once we have the exact solution, we can calculate the error at t = 1 by subtracting the numerical approximation obtained from the backward Euler method from the exact solution.
Let me pause here and calculate the values of y at each time step using the backward Euler method.