Kernel: SageMath 9.6

C09-C10 Numerical Methods

Exercise 1. Solve the logistic differential equation with a time varying carrying capacity u=u(1u1+0.25sin(2πt))u' = u\left(1 - \dfrac{u}{1 + 0.25\sin(2\pi t)}\right) and the initial condition u(0)=0.1u(0) = 0.1.

Exercise 2. Solve u=u3t2u' = u - 3t^2 with initial condition u(0)=1u(0) = 1, and find the values of u(0.25),u(0.50),u(0.75),u(1.00)u(0.25), u(0.50), u(0.75), u(1.00).

Exercise 3. Develop Euclid's Method. Use it to approximate the solution to u=u3t2,u(0)=1u' = u - 3t^2, u(0) = 1 at t=0.25,0.50,0.75,1.00t = 0.25, 0.50, 0.75, 1.00. Show the results graphically in comparison with the actual solution.

Exercise 4. Approximate u(1)u(1) for the initial value problem u=u3t2,u(0)=1u' = u - 3t^2, u(0) = 1 using Euclid's method with h=1,0.1,0.01,0.001,0.0001h = 1, 0.1, 0.01, 0.001, 0.0001 and compare with the actual value.

Exercise 5. Approximate u(1)u(1) for the initial value problem u=u3t2,u(0)=1u' = u - 3t^2, u(0) = 1 using the improved Euclid's method with h=1,0.1,0.01,0.001,0.0001h = 1, 0.1, 0.01, 0.001, 0.0001 and compare with the actual value.

Exercise 6. Approximate u(1)u(1) for the initial value problem u=u3t2,u(0)=1u' = u - 3t^2, u(0) = 1 using the fourth-order Runge-Kutta method with h=1,0.1,0.01,0.001,0.0001h = 1, 0.1, 0.01, 0.001, 0.0001 and compare with the actual value.