Consider the differential equation dy dx = sin(xy), with initial condition y(1) = 1

Using Eulers method, complete the following table for y = (x 2)(y 3)

Using Eulers method, complete the following table for y = 4xy.

A population, P, in millions, is 1500 at time t = 0 and its growth is governed by dP dt = 0.00008P(1900 P). Use Eulers method with t = 1 to estimate P at time t = 1, 2, 3.

(a) Use Eulers method to approximate the value of y at x = 1 on the solution curve to the differential equation dy/dx = 3 that passes through (0, 2). Use x = 0.2. (b) What is the solution to the differential equation dy/dx = 3 with initial condition y = 2 when x = 0? (c) What is the error for the Eulers method approximation at x = 1? (d) Explain why Eulers method is exact in this case

(a) Use five steps of Eulers method to determine an approximate solution for the differential equation dy/dx = yx with initial condition y(0) = 10, using step size x = 0.2. What is the estimated value of y at x = 1? (b) Does the solution to the differential equation appear to be concave up or concave down? (c) Are the approximate values overestimates or underestimates?

(a) Use ten steps of Eulers method to determine an approximate solution for the differential equation y = x3, y(0) = 0, using a step size x = 0.1. (b) What is the exact solution? Compare it to the computed approximation. (c) Use a sketch of the slope field for this equation to explain the results of part (b).

Consider the differential equation y = x + y whose slope field is in Figure 11.20 on page 595. Use Eulers method with x = 0.1 to estimate y when x = 0.4 for the solution curves satisfying (a) y(0) = 1 (b) y(1) = 0.

(a) Using Figure 11.38, sketch the solution curve that passes through (0, 0) for the differential equation dy dx = x3 y3 . (b) Compute the points on the solution curve generated by Eulers method with 5 steps of x = 0.2. (c) Is your answer to part (b) an overestimate or an underestimate?

Consider the solution of the differential equation y = y passing through y(0) = 1. (a) Sketch the slope field for this differential equation, and sketch the solution passing through the point (0, 1). (b) Use Eulers method with step size x = 0.1 to estimate the solution at x = 0.1, 0.2,..., 1. (c) Plot the estimated solution on the slope field; compare the solution and the slope field. (d) Check that y = ex is the solution of y = y with y(0) = 1.

Consider the differential equation y = (sin x)(sin y). (a) Calculate approximate y-values using Eulers method with three steps and x = 0.1, starting at each of the following points: (i) (0, 2) (ii) (0, ) (b) Use the slope field in Figure 11.25 on page 597 to explain your solution to part (a)(ii).

(a) Use Eulers method with five subintervals to approximate the solution curve to the differential equation dy/dx = x2 y2 passing through the point (0, 1) and ending at x = 1. (Keep the approximate function values to three decimal places.) (b) Repeat this computation using ten subintervals, again ending with x = 1.

Why are the approximate results you obtained in Problem 11 smaller than the true values? (Note: The slope field for this differential equation is one of those in Figure 11.18 on page 595.)

How do the errors of the five-step calculation and the tenstep calculation in Problem 11 compare? Estimate the true value of y on the solution through the point (0, 1) when x = 1.

(a) Use ten steps of Eulers method to approximate yvalues for dy/dt = 1/t, starting at (1, 0) and using t = 0.1. (b) Using integration, solve the differential equation to find the exact value of y at the end of these ten steps. (c) Is your approximate value of y at the end of ten steps bigger or smaller than the exact value? Use a slope field to explain your answer

Consider the differential equation dy dx = 2x, with initial condition y(0) = 1.

Consider the differential equation dy dx = sin(xy), with initial condition y(1) = 1. Estimate y(2), using Eulers method with step sizes x = 0.2, 0.1, 0.05. Plot the computed approximations for y(2) against x. What do you conclude? Use your observations to estimate the exact value of y(2

Use Eulers method to estimate B(1), given that dB dt = 0.05B and B = 1000 when t = 0. Take: (a) t = 1 and 1 step (b) t = 0.5 and 2 steps (c) t = 0.25 and 4 steps (d) Suppose B is the balance in a bank account earning interest. Explain why the result of your calculation in part (a) is equivalent to compounding the interest once a year instead of continuously. (e) Interpret the result of your calculations in parts (b) and (c) in terms of compound interest

Consider the differential equation dy/dx = f(x) with initial value y(0) = 0. Explain why using Eulers method to approximate the solution curve gives the same results as using left Riemann sums to approximate - x 0 f(t) dt.

Eulers method never produces an exact solution to a differential equation at a point. There is always some error.

If we use Eulers method on the interval [0, 1] to estimate the value of x(1) where dx/dt = x, then we get an underestimate

A differential equation for which the approximate values found using Eulers method lie on a straight line.

A differential equation and initial condition such that for any step size, the approximate y-value found after one step of Eulers method is an underestimate of the solution value.

Eulers method gives the arc length of a solution curve

Using Eulers method with five steps and x = 0.2 to approximate y(1) when dy/dx = f(x) and y(0) = 0 gives the same answer as the left Riemann sum approximation to - 1 0 f(x) dx.