Question: Match the differential equation with its direction
Match the differential equation with its direction field (labeled I–IV). Give reasons for your answer.
Transcribed Image Text:
y' = x + y – 1 III -2 حر ر
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> Suppose you have just poured a cup of freshly brewed coffee with temperature 95°C in a room where the temperature is 20°C. (a) When do you think the coffee cools most quickly? What happens to the rate of cooling as time goes by? Explain. (b) Newton’s Law
> Match the differential equations with the solution graphs labeled I–IV. Give reasons for your choices. (a) y' = 1 + x? + y² (b) у' — хе * y? 1 (c) y' (d) y' = sin(xy) cos(xy) 1 + e*²+y" I II III IV yA
> The function with the given graph is a solution of one of the following differential equations. Decide which is the correct equation and justify your answer. А. у'— 1 + ху B. y' = -2xy С. у'— 1 - 2ху
> Explain why the functions with the given graphs can’t be solutions of the differential equation dy e'(y – 1)? dt (а) Уд (b) У 1
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> (a) What can you say about the graph of a solution of the equation / when x is close to 0? What if x is large? (b) Verify that all members of the family / are solutions of the differential equation / (c) Graph several members of the family of solutions
> (a) What can you say about a solution of the equation y’ = -y2 just by looking at the differential equation? (b) Verify that all members of the family /are solutions of the equation in part (a). (c) Can you think of a solution of the di
> (a) Show that every member of the family of functions y = (In x + C)/x is a solution of the differential equation / (b) Illustrate part (a) by graphing several members of the family of solutions on a common screen. (c) Find a solution of the differential
> Which of the following functions are solutions of the differential equation / у" + у— sin x? SI
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> In Exercise 9.1.14 we considered a 958C cup of coffee in a 208C room. Suppose it is known that the coffee cools at a rate of 18C per minute when its temperature is 70°C. (a) What does the differential equation become in this case? (b) Sketch a direction
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> (a) Program your computer algebra system, using Euler’s method with step size 0.01, to calculate ys2d, where y is the solution of the initial-value problem (b) Check your work by using the CAS to draw the solution curve. y' = x' -
> Find the radius of convergence and interval of convergence of the series. n!x" Σ 1.3· 5. .... (2n – 1)
> (a) Program a calculator or computer to use Euler’s method to compute ys1d, where y(x) is the solution of the initial value problem (b) Verify that y = 2 + e2x3 is the exact solution of the differential equation. (c) Find the errors in
> (a) Use Euler’s method with step size 0.2 to estimate ys0.6d, where y(x) is the solution of the initial-value problem / (b) Repeat part (a) with step size 0.1.
> Use Euler’s method with step size 0.1 to estimate ys0.5d, where y(x) is the solution of the initial-value problem /
> Use Euler’s method with step size 0.2 to estimate ys1d, where y(x) is the solution of the initial-value problem /
> Use Euler’s method with step size 0.5 to compute the approximate y-values y1, y2, y3, and y4 of the solution of the initial-value problem /
> 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 overe
> (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 / / (b) We know that the exact solution of the initial-value problem in part (a) is y − ex. Draw, as accur
> Make a rough sketch of a direction field for the autonomous differential equation / where the graph off is as shown. How does the limiting behavior of solutions depend on the value of y(0)? y' = f(y).
> Use a computer algebra system to draw a direction field for the differential equation / 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 / What are the possible value
> Use a computer algebra system 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). Then use the CAS to draw the solution curve and compare it with your sketch. y
> Find the radius of convergence and interval of convergence of the series. x" Σ 1 1 · 3 · 5 · .. 1:3. 5. ... · (2n – 1)
> Use a computer algebra system 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). Then use the CAS to draw the solution curve and compare it with your sketch. y
> Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point y' - x + y', (0, 0)
> Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point. у' — у+ ху, (0, 1)
> Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point. У — ху — х, (0, 1)
> Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point. у — у — 2х, (1, 0)
> Sketch a direction field for the differential equation. Then use it to sketch three solution curves. - y' = x - y + 1
> Sketch a direction field for the differential equation. Then use it to sketch three solution curves. y' = ły
> Use the direction field labeled III (above) to sketch the graphs of the solutions that satisfy the given initial conditions. (b) у(0) — 2.5 (с) у(0) — 3.5 %3D (а) у(0) — 1 %3D %3D
> Use the direction field labeled I (above) to sketch the graphs of the solutions that satisfy the given initial conditions. (а) у(0) — 1 () у(0) — 2.5 (с) у(0) — 3.5
> Match the differential equation with its direction field (labeled I–IV). Give reasons for your answer. y' = sin x sin y yA IV -- -- -
> Find the radius of convergence and interval of convergence of the series. Σ n(In n)² n-2
> (a) For what values of k does the function y = cos k t satisfy the differential equation / (b) For those values of k, verify that every member of the family of functions y = A sin k t + B cos k t is also a solution.
> Match the differential equation with its direction field (labeled I–IV). Give reasons for your answer. y' = 2 - y yA I --- %3D -- 2 * // 2.
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> (a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylor’s Inequality to estimate the accuracy of the approximation / when x lies in the given interval. (c) Check your result in part (b) by graphing / S
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> A tank contains 1000 L of pure water. Brine that contains 0.05 kg of salt per liter of water enters the tank at a rate of 5 L/min. Brine that contains 0.04 kg of salt per liter of water enters the tank at a rate of 10 L/min. The solution is kept thorough
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> A tank contains 1000 L of brine with 15 kg of dissolved salt. Pure water enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank (a) after t minutes and (b) afte
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> In contrast to the situation of Exercise 40, experiments show that the reaction / satisfies the rate law and so for this reaction the differential equation becomes where x = [HBr] and a and b are the initial concentrations of hydrogen and bromine. (a) Fi
> In an elementary chemical reaction, single molecules of two reactants A and B form a molecule of the product C: A + B → C. The law of mass action states that the rate of reaction is proportional to the product of the concentrations of A
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> Solve the initial-value problem in Exercise 9.2.27 to find an expression for the charge at time t. Find the limiting value of the charge. Data from Exercise 9.2.27: The figure shows a circuit containing an electromotive force, a capacitor with a capaci
> Find a function f such that f(3) = 2 and (1? + 1)f"(1) + [f(1)]² + 1 = 0 t#1 ||
> An integral equation is an equation that contains an unknown function y(x) and an integral that involves y(x). Solve the given integral equation. y(x) у() — 4 + ("2у) 21/y(1) dt
> An integral equation is an equation that contains an unknown function y(x) and an integral that involves y(x). Solve the given integral equation. dt Уб) — 2 + ( ry(1) y(x) x>0
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> Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. y
> Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. y? = kx' .3
> Find the radius of convergence and interval of convergence of the series. E n!(2x – 1)"
> Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. x² + 2y? = k?
> (a) Use a computer algebra system to draw a direction field for the differential equation. Get a printout and use it to sketch some solution curves without solving the differential equation. (b) Solve the differential equation. (c) Use the CAS to draw se
> (a) Use a computer algebra system to draw a direction field for the differential equation. Get a printout and use it to sketch some solution curves without solving the differential equation. (b) Solve the differential equation. (c) Use the CAS to draw se
> Solve the differential equation y’ = x + y by making the change of variable u = x + y.
> Find the function f such that f’(x) = xf(x) – x and f(0) = 2.
> Find the radius of convergence and interval of convergence of the series. b" Σ -(х — а)", Ь>0 In n n-2
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> Find the solution of the differential equation that satisfies the given initial condition. у' tan x — а + у, у(п/3) — а, 0<x<п/2
> Find the solution of the differential equation that satisfies the given initial condition. dP Pt , dt Р(1) — 2
> Find the solution of the differential equation that satisfies the given initial condition. x In x = y(1 + /3 + y² ) y', y(1) = 1
> Find the solution of the differential equation that satisfies the given initial condition. dy x + 3y' /x? + T = 0, y(0) = 1 dx
> Find the solution of the differential equation that satisfies the given initial condition. du 21 + sec?t u(0) = -5 %3D dt 2u
> Find the solution of the differential equation that satisfies the given initial condition. dy x sin x y(0) dx y
> Find the solution of the differential equation that satisfies the given initial condition. dy - хе', у(0) — 0 dx
> Solve the differential equation. dz + ettz = 0 dt
> Find the radius of convergence and interval of convergence of the series. - (х — а)", Ь> 0 Σ b" n-1
> Solve the differential equation. dp uP. = t°p – p + t² – 1 dt
> Solve the differential equation. dH_ /T + R² In H RH² dR
> Solve the differential equation. de t sec 0 dt Oe“
