13. Intro to Differential Equations

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

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. 

Use Euler’s method with dx = 1/3 to estimate y(2) if y′ = x sin y and y(0) = 1. What is the exact value of y(2)?

Using Euler’s Method

In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.

y' = 2xy + 2y, y(0) = 3, dx = 0.2

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) = -2y, y(0) = 1; Δt = 0.2, T = 2; y(t) = e⁻²ᵗ

Use Euler’s method with dx = 0.5 to estimate y(5) if y′ = y²/√x and y(1) = −1. What is the exact value of y(5)?

Use Euler’s method with a step size of $h=0.5$ to estimate the value of $y\left(2\right)$, where $y$ is the solution of the initial value problem $y^{\prime }=2x,y\left(0\right)=1$

Euler’s method on more general grids Suppose the solution of the initial value problem y'(t)=f(t,y),y(a)=A is to be approximated on the interval [a, b].

b. Write the first step of Euler’s method to compute u1.

Euler’s method Consider the initial value problem y′(t)=1/2y,y(0)=1. 

b. Use Euler’s method with Δt=0.05 to compute approximations to y(0.1) and y(0.2). 

Stability of Euler's method Consider the initial value problem y′(t) = −ay, y(0) = 1 where a > 0; it has the exact solution y(t) = e⁻ᵃᵗ, which is a decreasing function.

a. Show that Euler's method applied to this problem with time step h can be written u₀ = 1, uₖ₊₁ = (1 − ah)uₖ for k = 0, 1, 2, ...

b. Show by substitution that uₖ = (1 − ah)ᵏ is a solution of the equations in part (a), for k = 0, 1, 2, ...

Convergence of Euler's method Suppose Euler's method is applied to the initial value problem y′(t) = ay, y(0) = 1, which has the exact solution y(t) = eᵃᵗ. For this exercise, let h denote the time step (rather than Δt). The grid points are then given by tₖ = kh. We let uₖ be the Euler approximation to the exact solution y(tₖ), for k = 0, 1, 2, ...

b. Show by substitution that uₖ = (1 + ah)ᵏ is a solution of the equations in part (a), for k = 0, 1, 2, ...

Using Euler’s Method

In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.

y' = 2y/x, y(1) = -1, dx = 0.5

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

Using Euler’s Method

In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.

y' = y²(1+2x), (y-1) = 1, dx = 0.5