Back33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.
b. Using the exact solution (also given), find the error in the approximation to y(T) (only at the right endpoint of the time interval).
y′(t) = -2y, y(0) = 1; Δt = 0.2, T = 2; y(t) = e⁻²ᵗ
Use Euler’s method with a step size of to estimate the value of , where is the solution of the initial value problem
Let with . Compute the first three approximations given by Euler’s Method with a step size of .
25–28. Two steps of Euler’s method For the following initial value problems, compute the first two approximations u1 and u2 given by Euler’s method using the given time step.
y′(t) = 2−y, y(0) = 1; Δt = 0.1
29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.
b. Using the exact solution given, compute the errors in the Euler approximations at t=0.2 and t=0.4.
y′(t) = −y, y(0) = 1; y(t) = e⁻ᵗ
29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.
a. Find the approximations to y(0.2) and y(0.4) using Euler’s method with time steps of Δt = 0.2, 0.1, 0.05, and 0.025.
y′(t) = y/2, y(0) = 2; y(t) = 2eᵗᐟ²
29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.
d. In general, how does halving the time step affect the error at t=0.2 and t=0.4?
y′(t) = y/2, y(0) = 2; y(t) = 2eᵗᐟ²
29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.
c. Which time step results in the more accurate approximation? Explain your observations.
y′(t) = 4−y, y(0) = 3; y(t) = 4−e⁻ᵗ
29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.
b. Using the exact solution given, compute the errors in the Euler approximations at t=0.2 and t=0.4.
y′(t) = 2t + 1, y(0) = 0; y(t) = t² + t
33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.
a. Approximate the value of y(T) using Euler’s method with the given time step on the interval [0,T].
y′(t) = -2y, y(0) = 1; Δt = 0.2, T = 2; y(t) = e⁻²ᵗ
33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.
c. Repeating parts (a) and (b) using half the time step used in those calculations, again find an approximation to y(T).
y′(t) = 6 - 2y, y(0) = -1; Δt = 0.2, T = 3; y(t) = 3 - 4e⁻²ᵗ
33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.
d. Compare the errors in the approximations to y(T).
y′(t) = 6 - 2y, y(0) = -1; Δt = 0.2, T = 3; y(t) = 3 - 4e⁻²ᵗ
33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.
a. Approximate the value of y(T) using Euler’s method with the given time step on the interval [0,T].
y′(t) = t/y, y(0) = 4; Δt = 0.1, T = 2; y(t) = √(t² + 16)
33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.
b. Using the exact solution (also given), find the error in the approximation to y(T) (only at the right endpoint of the time interval).
y′(t) = t/y, y(0) = 4; Δt = 0.1, T = 2; y(t) = √(t² + 16)
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. Euler’s method is used to compute exact values of the solution of an initial value problem.