?Use a computer to draw a direction field for the given differential equation. Get a

A direction field for the differential equation \(y^{\prime}=x \cos \pi y\) is shown. (a) Sketch the graphs of the solutions that satisfy the given initial conditions. (i) \(y(0)=0\) (ii) \(y(0)=0.5\) (iii) \(y(0)=1\) (iv) \(y(0)=1.6\) (b) Find all the equilibrium solutions. Equation Transcription: Text Transcription: y'= x cos pi y y(0) = 0 y(0) = 0.5 y(0) = 1 y(0) = 1.6

A direction field for the differential equation \(y^{\prime}=\tan \left(\frac{1}{2} \pi y\right)\) is shown. (a) Sketch the graphs of the solutions that satisfy the given initial conditions. (i) \(y(0)=1\) (ii) \(y(0)=0.2\) (iii) \(y(0)=2\) (iv) \(y(1)=3\) (b) Find all the equilibrium solutions. Equation Transcription: Text Transcription: y'=tan(1/2 pi y) y(0) = 1 y(0) = 0.2 y(0) = 2 y(1) = 3

Match the differential equation with its direction field (labeled I–IV). Give reasons for your answer. \(y^{\prime}=2-y\) Equation Transcription: Text Transcription: y’ = 2 ? y

Match the differential equation with its direction field (labeled I–IV). Give reasons for your answer. \(y^{\prime}=x(2-y)\) Equation Transcription: Text Transcription: y’ = x(2 ? y)

Match the differential equation with its direction field (labeled I–IV). Give reasons for your answer. \(y^{\prime}=x+y-1\) Equation Transcription: Text Transcription: y’ = x + y ? 1

Match the differential equation with its direction field (labeled I-IV). Give reasons for your answer. \(y^{\prime}=\sin x \sin y\)

Use the direction field labeled I (above) to sketch the graphs of the solutions that satisfy the given initial conditions. (a) \(y(0) = 1\) (b) \(y(0) = 2.5\) (c) \(y(0) = 3.5\) Equation Transcription: Text Transcription: y(0)=1 y(0)=2.5 y(0)=3.5

Use the direction field labeled III (above) to sketch the graphs of the solutions that satisfy the given initial conditions. (a) \(y(0) = 1\) (b) \(y(0) = 2.5\) (c) \(y(0) = 3.5\) Equation Transcription: Text Transcription: y(0)=1 y(0)=2.5 y(0)=3.5

Sketch a direction field for the differential equation. Then use it to sketch three solution curves. \(y^{\prime}=\frac{1}{2} y\) Equation Transcription: Text Transcription: y'=1/2 y

Sketch a direction field for the differential equation. Then use it to sketch three solution curves. \(y^{\prime}=x-y+1\) Equation Transcription: Text Transcription: y'=x-y+1

Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point.

Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point.

Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point. \(y^{\prime}=y+x y\), (0, 1)

Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point.

Use a computer to draw a direction field for the given differential equation. Get a printout and sketch on it the solution curve that passes through (0, 1). Compare your sketch to a computer-drawn solution curve.

Use a computer to draw a direction field for the given differential equation. Get a printout and sketch on it the solution curve that passes through (0, 1). Compare your sketch to a computer-drawn solution curve.

Use a computer to draw a direction field for the differential equation y’ = y3 ? 4y. Get a printout and sketch on it solutions that satisfy the initial condition y(0) = c for various values of c. For what values of c does y(t) exist? What are the possible values for this limit?

Make a rough sketch of a direction field for the autonomous differential equation y’ = f(y), where the graph of f is as shown. How does the limiting behavior of solutions depend on the value of y(0)?

(a) Use Euler’s method with each of the following step sizes to estimate the value of y(0.4), where y is the solution of the initial-value problem y’ = y, y(0) = 1. (i) h = 0.4 (ii) h = 0.2 (iii) h = 0.1 (b) We know that the exact solution of the initial-value problem in part (a) is y = ex. Draw, as accurately as you can, the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>e</mi><mi>x</mi></msup><mo>,</mo><mn>0</mn><mo>&#x2A7D;</mo><mi>x</mi><mo>&#x2A7D;</mo><mn>0</mn><mo>.</mo><mn>4</mn></math> , together with the Euler approximations using the step sizes in part (a). (Your sketches should resemble Figures 11, 12, and 13.) Use your sketches to decide whether your estimates in part (a) are underestimates or overestimates. (c) The error in Euler’s method is the difference between the exact value and the approximate value. Find the errors made in part (a) in using Euler’s method to estimate the true value of y(0.4), namely, e0.4. What happens to the error each time the step size is halved?

A direction field for a differential equation is shown. Draw, with a ruler, the graphs of the Euler approximations to the solution curve that passes through the origin. Use step sizes h = 1 and h = 0.5. Will the Euler estimates be underestimates or overestimates? Explain.

Use Euler's method with step size 0.5 to compute the approximate y-values \(y_{1}, y_{2}, y_{3}\), and \(y_{4}\) of the solution of the initial-value problem \(y^{\prime}=y-2 x, \ y(1)=0\).

Use Euler’s method with step size 0.2 to estimate y(1), where y(x) is the solution of the initial-value problem y’ = x2y ? y2, y(0) = 1.

Use Euler’s method with step size 0.1 to estimate y(0.5), where y(x) is the solution of the initial-value problem y’ = y + xy, y(0) = 1.

(a) Use Euler’s method with step size 0.2 to estimate y(0.6), where y(x) is the solution of the initial-value problem y’ = cos(x + y), y(0) = 0. (b) Repeat part (a) with step size 0.1.

(a) Program a calculator or computer to use Euler’s method to compute y(1), where y(x) is the solution of the initial-value problem y(0) = 3 (i) h = 1 (ii) h = 0.1 (iii) h = 0.01 (iv) h = 0.001 (b) Verify that is the exact solution of the differential equation. (c) Find the errors in using Euler’s method to compute y(1) with the step sizes in part (a). What happens to the error when the step size is divided by 10?

(a) Use Euler’s method with step size 0.01 to calculate y(2), where y is the solution of the initial-value problem y’ = x3 ? y3 y(0) = 1 (b) Compare your answer to part (a) to the value of y(2) that appears on a computer-drawn solution curve.

The figure shows a circuit containing an electromotive force, a capacitor with a capacitance of C farads (F), and a resistor with a resistance of R ohms (?). The voltage drop across the capacitor is Q/C, where Q is the charge (in coulombs, C), so in this case Kirchhoff’s Law gives But I = dQ/dt, so we have Suppose the resistance is 5 ?, the capacitance is 0.05 F, and a battery gives a constant voltage of 60 V. (a) Draw a direction field for this differential equation. (b) What is the limiting value of the charge? (c) Is there an equilibrium solution? (d) If the initial charge is Q(0) = 0 C, use the direction field to sketch the solution curve. (e) If the initial charge is Q(0) = 0 C, use Euler’s method with step size 0.1 to estimate the charge after half a second.

In Exercise 9.1.26 we considered a 95°C cup of coffee in a 20°C room. Suppose it is known that the coffee cools at a rate of 1°C per minute when its temperature is 70°C. (a) What does the differential equation become in this case? (b) Sketch a direction field and use it to sketch the solution curve for the initial-value problem. What is the limiting value of the temperature? (c) Use Euler’s method with step size h = 2 minutes to estimate the temperature of the coffee after 10 minutes.