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Reproduce the given computer-generated direction fie­ld. Then sketch, by hand, an approximate solution curve that passes through each of the indicated points. Use different colored pencils for each solution curve.


1. dy/dx = x2 –y2

(a) y(-2) = 1

(b) y(3) = 0

(c) y(0) = 2

(d) y(0) = 0

 

2. dy/dx = e-0.01xy2

(a) y(-6) = 0

(b) y(0) = 1

(c) y(0) = -4

(d) y(8) = -4


3. dy/dx = 1 – xy

(a) y(0) = 0

(b) y(-1) = 0

(c) y(2) = 2

(d) y(0) = -4


4. dy/dx = (sinx)cosy

(a) y(0) = 1

(b) y(1) = 0

(c) y(3) = 3

(d) y(d) = -5/2

Use computer software to obtain a direction ­field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points.

1. y' = x

(a) y(0) = 0

(b) (-2)

2. y' = x + y

(a) y(-2) = 2

(b) y(1) -3

3. y dy/dx = -x

(a) y(1) = 1

(b) y(0) = 4

4. dy/dx = 1/y

(a) y(0) = 1

(b) y(-2) = -1

5. dy/dx = 0.2x2 + y

(a) y(0) = ½

(b) y(2) = -2

6. dy/dx = xex

(a) y(0) = -2

(b) y(1) = 2.5

7. y' = y – cos π/2 x

(a) y(2) = 2

(b) y(-1) = 0

8. dy/dx = 1 – y/x

(a) y(-1/2) = 2

(b) y(3/2) = 0

The given fi­gure represents the graph of f (y) and f (x), respectively. By hand, sketch a direction  field over an appropriate grid for dy/dx = f (y) (Problem 13) and then for dy/dx = f (x) (Problem 14).

1.

 


2.

In parts (a) and (b) sketch isoclines f (x, y) = c for the given differential equation using the indicated values of c. Construct a direction ­field over a grid by carefully drawing lineal elements with the appropriate slope at chosen points on each isocline. In each case, use this rough direction fi­eld to sketch an approximate solution curve for the IVP consisting of the DE and the initial condition y(0) = 1.

(a) dy/dx = x + y; c an integer satisfying -5 ≤ c ≤ 5

(b) dy/dx = x2 + y2; c = 1/4, c = 1, c = 9/4, c = 4


(a) Consider the direction fi­eld of the differential equation dy/dx = x(y - 4)2 - 2, but do not use technology to obtain it. Describe the slopes of the lineal elements on the lines x = 0, y = 3, y = 4, and y = 5.

(b) Consider the IVP dy/dx = x(y - 4)2 - 2, y(0) = y0, where y0 < 4. Can a solution y(x) → ∞? Based on the information in part (a), discuss.

For a fi­rst-order DE dy/dx = f (x, y) a curve in the plane defi­ned by f (x, y) = 0 is called a nullcline of the equation, since a lineal element at a point on the curve has zero slope. Use computer software to obtain a direction ­field over a rectangular grid of points for dy/dx = x2 - 2y, and then superimpose the graph of the nullcline y = 1/2 x2 over the direction fi­eld. Discuss the behavior of solution curves in regions of the plane defi­ned by y < ½ x2 and by y > ½ x2. Sketch some approximate solution curves. Try to generalize your observations.

(a) Identify the nullclines (see Problem 17) in Problems‑1, 3, and 4. With a colored pencil, circle any lineal elements in figures (1), (2) and (3) that you think may be a lineal element at a point on a nullcline.

(b) What are the nullclines of an autonomous fi­rst-order DE?

(1)

 

(2)

(3)

Data from problem 17

For a fi­rst-order DE dy/dx = f (x, y) a curve in the plane defi­ned by f (x, y) = 0 is called a nullcline of the equation, since a lineal element at a point on the curve has zero slope. Use computer software to obtain a direction ­field over a rectangular grid of points for dy/dx = x2 - 2y, and then superimpose the graph of the nullcline y = 1/2 x2 over the direction fi­eld. Discuss the behavior of solution curves in regions of the plane defi­ned by y < ½ x2 and by y > ½ x2. Sketch some approximate solution curves. Try to generalize your observations.

Consider the autonomous fi­rst-order differential equation dy/dx = y - y3 and the initial condition y(0) = y0. By hand, sketch the graph of a typical solution y(x) when y0 has the given values.

(a) y0 > 1

(b) 0 < y0 < 1

(c) -1 < y0 < 0

(d) y0 < -1

Consider the autonomous ­rst-order differential equation dy/dx = y2 - y4 and the initial condition y(0) = y0. By hand, sketch the graph of a typical solution y(x) when y0 has the given values.

(a) y0 > 1

(b) 0 < y0 < 1

(c) -1 < y0 < 0

(d) y0 < -1

Fi­nd the critical points and phase portrait of the given autonomous fi­rst-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions.

1. dy/dx = y2 – 3y

2. dy/dx = y2 – y3

3. dy/dx = (y - 2)4

4. dy/dx = 10 + 3y – y2

5. dy/dx = y2(4 – y2)

6. dy/dx = y(2 - y)(4 – y)

7. dy/dx = y ln(y + 2)

8. dy/dx = (yey – 9y) / ey

Consider the autonomous differential equation dy/dx = f (y), where the graph of f is given. Use the graph to locate the critical points of each differential equation. Sketch a phase portrait of each differential equation. By hand, sketch typical solution curves in the subregions in the xy-plane determined by the graphs of the equilibrium solutions.

1.

 


2.

Consider the autonomous DE dy/dx = (2/πy­)y - sin y. Determine the critical points of the equation. Discuss a way of obtaining a phase portrait of the equation. Classify the critical points as asymptotically stable, unstable, or semi-stable.

A critical point c of an autonomous ­first-order DE is said to be isolated if there exists some open interval that contains c but no other critical point. Can there exist an autonomous DE of the form given in (2) for which every critical point is nonisolated? Discuss; do not think profound thoughts.

Suppose that y(x) is a nonconstant solution of the autonomous equation dy/dx = f (y) and that c is a critical point of the DE. Discuss: Why can’t the graph of y(x) cross the graph of the equilibrium solution y = c? Why can’t f (y) change signs in one of the subregions discussed on page 40? Why can’t y(x) be oscillatory or have a relative extremum (maximum or minimum)?

Suppose that y(x) is a solution of the autonomous equation  dy/dx = f (y) and is bounded above and below by two consecutive critical points c1< c2, as in subregion R2of  the following figure. If f(y) > 0 in the region, then limx → ∞ y(x) = c2. Discuss why there cannot exist a number L < c2such that limx →∞ y(x) = L. As part of your discussion, consider what happens to y'(x) as x →∞

 

Using the autonomous equation (2), discuss how it is possible to obtain information about the location of points of inflection of a solution curve.

Consider the autonomous DE dy/dx = y2 - y - 6.Use your ideas from Problem 35 to ­find intervals on the y-axis for which solution curves are concave up and intervals for which solution curves are concave down. Discuss why each solution curve of an initial-value problem of the form dy/dx = y2 - y - 6, y(0) = y0, where -2 < y0 < 3, has a point of inflection with the same y coordinate. What is that y-coordinate? Carefully sketch the solution curve for which y(0) = -1. Repeat for y(2) = 2.


Data from problem 35

Using the autonomous equation (2), discuss how it is possible to obtain information about the location of points of inflection of a solution curve.

Suppose the autonomous DE in (2) has no critical points. Discuss the behavior of the solutions.

Population Model The differential equation in Example 3 is a well-known population model. Suppose the DE is changed to

dP/dt = P(aP - b),

where a and b are positive constants. Discuss what happens to the population P as time t increases.

Population Model Another population model is given by

dP/dt = kP - h,

where h and k are positive constants. For what initial values P(0) = P0 does this model predict that the population will go extinct?

Terminal Velocity In Section 1.3 we saw that the autonomous differential equation

M dv/dt = mg - kv,

where k is a positive constant and g is the acceleration due to gravity, is a model for the velocity v of a body of mass m that is falling under the influence of gravity. Because the term 2kv represents air resistance, the velocity of a body falling from a great height does not increase without bound as time t increases. Use a phase portrait of the differential equation to fi­nd the limiting, or terminal, velocity of the body. Explain your reasoning.

Suppose the model in Problem 40 is modi­ed so that air resistance is proportional to v2, that is,

M dv/dt = mg - kv2.

See Problem 17 in Exercises 1.3. Use a phase portrait to fi­nd the terminal velocity of the body. Explain your reasoning.


Data from problem 17

For a fi­rst-order DE dy/dx = f (x, y) a curve in the plane defi­ned by f (x, y) = 0 is called a nullcline of the equation, since a lineal element at a point on the curve has zero slope. Use computer software to obtain a direction ­field over a rectangular grid of points for dy/dx = x2 - 2y, and then superimpose the graph of the nullcline y = ½ x2 over the direction fi­eld. Discuss the behavior of solution curves in regions of the plane de­ned by y < ½ x2 and by y > ½ x2. Sketch some approximate solution curves. Try to generalize your observations.

Data from problem 40

Terminal Velocity In Section 1.3 we saw that the autonomous differential equation

M dv/dt = mg - kv,

where k is a positive constant and g is the acceleration due to gravity, is a model for the velocity v of a body of mass m that is falling under the influence of gravity. Because the term 2kv represents air resistance, the velocity of a body falling from a great height does not increase without bound as time t increases. Use a phase portrait of the differential equation to fi­nd the limiting, or terminal, velocity of the body. Explain your reasoning.

Chemical Reactions When certain kinds of chemicals are combined, the rate at which the new compound is formed is modeled by the autonomous differential equation

dX/dt = k(α - X)(β - X),

where k . 0 is a constant of proportionality and β > α > 0. Here X(t) denotes the number of grams of the new compound formed in time t.

(a) Use a phase portrait of the differential equation to predict the behavior of X(t) as t → ∞

(b) Consider the case when α = β. Use a phase portrait of the differential equation to predict the behavior of X(t) as t → ∞ when X(0) < α. When X(0) > α.

(c) Verify that an explicit solution of the DE in the case when k = 1 and α = β is X(t) = α – 1/(t + c). Find a solution that satisfi­es X(0) = α/2. Then fi­nd a solution that satisfi­es X(0) = 2α. Graph these two solutions. Does the behavior of the solutions as t → ∞ agree with your answers to part (b)?

Solve the given differential equation by separation of variables.

1. dy/dx = sinx

2. dy/dx = (x + 1)2

3. dx + e2x dy = 0

4. dy – (y - 1)2 dx = 0

5. x dy/dx = 4y

6. dy/dx + 2xy2 = 0

7. dy/dx = e3x+2y

8. ex ydy/dx = e-y + e-2x-y

9. y ln xdy/dx = (y + 1 / x)2

10. dy/dx = (2y + 3 / 4x + 5)2

11. csc ydx + secx dy = 0

12. sin3x dx + 2y cos33x dy = 0

13. (ey + 1)2 e-y dx + (ex + 1)3 e-x dy = 0

14. x(1 + y2)1/2 dx = y(1 + x2)1/2 dy

15. dS/dr = kS

Fi­nd an explicit solution of the given initial-value problem.

1. dx/dt = 4(x2 + 2), x(π/4) =1

2. dy/dx = (y2 - 2)/ (x2 - 1), y(2) = 2

3. x2 dy/dx = y – xy, y(-1) = -1

4. dy/dt + 2y = 1, y(0) = 5/2

5. √(1 – y2) dx – √(1 – x2) dy = 0, y(0) = √3 / 2

6. √(1 – y4) dy + x(1 + 4y2) dx = 0, y(1) = 0

Proceed as in Example 5 and fi­nd an explicit solution of the given initial-value problem.

1. dy/dx = ye-x2, y(4) = 1

2. dy/dx = y2 sinx2, y(-2) = 1/3

Fi­nd an explicit solution of the given initial-value problem. Determine the exact interval I of definition by analytical methods. Use a graphing utility to plot the graph of the solution.

1. dy/dx = 2x + 1 / 2y, y(-2) = -1

2. (2y - 2)dy/dx 2x2 + 4x + 2, y(1) = -2

3. ey dx – e-x dy = 0, y(0) = 0

4. sinx dx + y dy = 0, y(0) = 1

(a) Find a solution of the initial-value problem consisting of the differential equation in Example 3 and each of the initial conditions:

y(0) = 2, y(0) = -2, and y(1/4) + = 1.

(b) Find the solution of the differential equation in Example 3 when ln c1 is used as the constant of integration on the left-hand side in the solution and 4 ln c1 is replaced by ln c. Then solve the same initial-value problems in part (a).

Find a solution of x dy/dx = y2 - y that passes through the indicated points.

(a) (0, 1)

(b) (0, 0)

(c) (12, ½)

(d) (2, ¼)

Find a singular solution of Problem 21. Of Problem 22.

Data from problem 21 & 22

21. dy/dx = x√(1 – y2)

22. (ex + e-x)dy/dx = y2

State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6).

1.

2.

3. t5y(4) t3y" + 6y = 0

4.

5.

6. d2R/dt2 = k/R2

7. (sinθ )ym (cosθ)y' = 2

8.

Determine whether the given first-order differential equation is linear in the indicated dependent variable by matching it with the first differential equation given in (7).

1. (y2  1) dx + x dy  0; in y; in x

2. u dv + (v + uv  ueu) du  0; in v; in u

Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution

1. 2y' + y = 0; y = e-x/2

2. dy/dt + 20y = 24; y – 6/5 – 6/5 e-20t

3. y'' – 6y' + 13y = 0; y = e3x cos 2x

4. y'' + y = tan x; y = -(cos x) ln (sec x + tan x)

verify that the indicated function y = Φ(x) is an explicit solution of the given first-orde differential equation. Proceed as in Example 2, by considering Φ simply as a function, give its domain. Then by considering Φ as a solution of the differential equation, give at least one interval I of definition

1. (y - x)y' = y – x + 8; y = x + 4 √(x + 2)

2. y' = 25 + y2; y = 5 tan 5x

3. y' = 2xy2; y = 1/(4 – x2)

4. 2y' = y3 cosx; y = (1 = sinx)-1/2

verify that the indicated expression is an implicit solution of the given first-order differential equation. Find at least one explicit solution y Φ (x) in each case.

Use a graphing utility to obtain the graph of an explicit solution. Give an interval I of definition of each solution Φ.

1.

 

2. 2xy dx + (x2 - y) dy = 0; -2x2y + y2 = 1

Verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution

1.

 

2.

3.

4.

Verify that the piecewise-defined functio


is a solution of the differential equation xy'- 2y = 0 on (-∞, ∞).

In Example 5 we saw that y = Φ1 (x) = √(25 – x2) and y =2(x) = -√(25 – x2) are solutions of dy/dx = -x/y on the interval (-5, 5). Explain why the peicewise-defined function

In Problems 31 and 32 find values of m so that the function y = xm is a solution of the given differential equation.

1. xy'' + 2y' = 0

2. x2y''  - 7xy' + 15y = 0

In Problems 1 and 2 verify that the indicated pair of functions is a solution of the given system of differential equations on the interval (-∞, ∞).

1.


2.

Make up a differential equation that does not possess any real solutions.

The fi­gure represents a portion of a direction fi­eld of an autonomous ­first-order differential equation dy/dx = f (y). Reproduce the fi­gure on a separate piece of paper and then complete the direction fi­eld over the grid. The points of the grid are (mh, nh), where h = 1/2, m and n integers, -7 ≤ m ≤ 7, -7 ≤ n ≤ 7. In each direction fi­eld, sketch by hand an approximate solution curve that passes through each of the solid points shown in red. Discuss: Does it appear that the DE possesses critical points in the interval -3.5 ≤ y ≤ 3.5? If so, classify the critical points as asymptotically stable, unstable, or semi-stable.

Use Euler’s method with step size h = 0.1 to approximate y(1.2), where y(x) is a solution of the initial-value problem y' = 1 + x√y, y(1) = 9.

The fi­gure represents a portion of a direction fi­eld of an autonomous ­first-order differential equation dy/dx = f (y). Reproduce the fi­gure on a separate piece of paper and then complete the direction fi­eld over the grid. The points of the grid are (mh, nh), where h = 1/2, m and n integers, -7 ≤ m ≤ 7, -7 ≤ n ≤ 7. In each direction fi­eld, sketch by hand an approximate solution curve that passes through each of the solid points shown in red. Discuss: Does it appear that the DE possesses critical points in the interval -3.5 ≤ y ≤ 3.5? If so, classify the critical points as asymptotically stable, unstable, or semi-stable.

Graphs of some members of a family of solutions for a fi­rst-order differential equation dy/dx = f (x, y) are shown in the following figure. The graphs of two implicit solutions, one that passes through the point (1, 21)‑and one that passes through (-1, 3), are shown in blue. Reproduce the ­figure on a piece of paper. With colored pencils trace out the solution curves for the solutions y = y1(x) and y = y2(x) de­fined by the implicit solutions such that y1(1) = -1 and y2(-1) = 3, respectively. Estimate the intervals on which the solutions y = y1(x) and y = y2(x) are de­fined.

(a) Find an implicit solution of the initial-value problem

dy/dx = (y2 - x2)/xy, y(1) = -√2.

(b) Find an explicit solution of the problem in part (a) and give the largest interval I over which the solution is defi­ned. A graphing utility may be helpful here.

Express the solution of the given initial-value problem in terms of an integral defi­ned function.

x dy/dx + (sin x)y = 0, y(0) = 10

Solve the given initial-value problem.

dy/dx + y = f (x), y(0) = 5, where

Solve the given initial-value problem.

dy/dx + P(x)y = ex, y(0) = -1, where

Solve the given initial-value problem and give the largest interval I on which the solution is defined.

sin x dy/dx + (cos x)y = 0, y(7π/y6) = -2

Solve the given initial-value problem and give the largest interval I on which the solution is defined.

dy/dt + 2(t + 1)y2 = 0, y(0) = -1/8

(a) Without solving, explain why the initial-value problem dy/dx = √y, y(x0) = yhas no solution for y0 , 0.

(b) Solve the initial-value problem in part (a) for y0 < 0 and fi­nd the largest interval I on which the solution is defi­ned.

In Problem solve the given differential equation

(2r2 cos θ sin θ + r cos θ) dθ + (4r + sin θ - 2r cos 2 θ) dr = 0 

Solve the given differential equation.

(x2 + 4) dy = (2x - 8xy) dx

Express the solution of the given initial-value problem in terms of an integral defi­ned function.

2 dy/dx + (4 cos x)y = x, y(0) = 1

Solve the given differential equation.

(2x + y + 1)y' = 1

Express the solution of the given initial-value problem in terms of an integral defi­ned function.

dy/dx - 4xy = sin x2, y(0) = 7

Solve the given differential equation.

t dQ/dt + Q = t4 ln t

Classify each differential equation as separable, exact, linear, homogeneous, or Bernoulli. Some equations may be more than one kind. Do not solve.

(a) dy/dx = x - y/x

(b) dy/dx = 1/y – x

(c) (x + 1) dy/dx = 2y + 10

(d) dy/dx = 1/x(x - y)

(e) dy/dx = y2 + y/x2 + x

(f) dy/dx = 5y + y2

(g) y dx = (y - xy2) dy

(h) x dy/dx = yex/y - x

(i) xy y' + y2 = 2x

(j) 2xy y' + y2 = 2x2

(k) y dx + x dy = 0

(l) (x2 + 2y/x) dx = (3 - ln x2) dy

(m) dy/dx = x/y + y/x + 1

(n) y/xdy/dx + e2x3+y2 = 0

The following figure is a portion of a direction ­field of a differential equation dy/dx = f (x, y). By hand, sketch two different solution curves—one that is tangent to the lineal element shown in black and one that is tangent to the lineal element shown in red.

Consider the differential equation dP/dt = f (P), where f (P) = 20.5P3- 1.7P + 3.4.

The function f (P) has one real zero, as shown in the following figure. Without attempting to solve the differential equation, estimate the value of limt→∞ P(t).

Solve the given differential equation.

dx/dy = - 4y2 + 6xy/3y2 + 2x

Solve the given differential equation.

(6x + 1)y2 dy/dx + 3x2 + 2y3 = 0

Solve the given differential equation.

y(ln x - ln y) dx = (x ln x - x ln y - y) dy

Solve the given differential equation.

(y2 + 1 dx) = y sec2 xdy

The number 0 is a critical point of the autonomous differential equation dx/dt = xn, where n is a positive integer. For what values of n is 0 asymptotically stable? Semi-stable? Unstable? Repeat for the differential equation dx/dt = 2xn.

Construct an autonomous ­first order differential equation dy/dx = f (y) whose phase portrait is consistent with the given ­figure.

Construct an autonomous ­first order differential equation dy/dx = f (y) whose phase portrait is consistent with the given ­figure.

Fill in the blanks or answer true or false.


is a solution of the linear ­first order differential equation ___________.

If a differentiable function y(x) satisfies y' = |x|, y(-1) = 2, then y(x) = ________________.

Fill in the blanks or answer true or false.

If y' = exy, then y = __________.

Fill in the blanks or answer true or false.


By inspection, two solutions of the differential equation y' + |y| = 2 are _______

Fill in the blanks or answer true or false.


Every autonomous DE dy/dx = f (y) is separable.________


Fill in the blanks or answer true or false.


The fi­rst-order DE, dr/dθ = rθ + r + θ + 1 is not separable.______

Fill in the blanks or answer true or false.


An example of a nonlinear third-order differential equation in normal form is _____

Fill in the blanks or answer true or false.


The linear DE, a1(x)y' + a0(x)y = 0 is also separable.­­­______

Fill in the blanks or answer true or false.


The linear DE, y' + k1y = k2, where k1 and k2 are nonzero constants, always possesses a constant solution.________

Fill in the blanks or answer true or false.


The initial-value problem x dy/dx - 4y = 0, y(0) = k, has an infi­nite number of solutions for k =______ and no solution for k = _____ .

Fill in the blanks or answer true or false.


The linear DE, y' - ky = A, where k and A are constants, is autonomous. The critical point of the equation is a(n) (attractor or repeller) for k > 0 and a(n) (attractor or repeller) for k < 0.

(a) Use a numerical solver and the RK4 method to graph the solution of the initial-value problem y' = -2xy + 1, (0) = 0.

(b) Solve the initial-value problem by one of the analytic procedures developed earlier in this chapter.

(c) Use the analytic solution y(x) found in part (b) and a CAS to fi­nd the coordinates of all relative extrema.

Use a numerical solver and Euler’s method to approximate y(1.0), where y(x) is the solution to y' = 2xy2, y(0) = 1. First use h = 0.1 and then use h = 0.05. Repeat, using the RK4 method. Discuss what might cause the approximations to y(1.0) to differ so greatly.


Use a numerical solver to obtain a numerical solution curve for the given initial-value problem. First use Euler’s method and then the RK4 method. Use h = 0.25 in each case. Superimpose both solution curves on the same coordinate axes. If possible, use a different color for each curve. Repeat, using h = 0.1 and h = 0.05.

y' = y(10 - 2y), y(0) = 1

Use a numerical solver to obtain a numerical solution curve for the given initial-value problem. First use Euler’s method and then the RK4 method. Use h = 0.25 in each case. Superimpose both solution curves on the same coordinate axes. If possible, use a different color for each curve. Repeat, using h = 0.1 and h = 0.05.

y' = 2(cos x)y, y(0) = 1

Use a numerical solver and Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05.

y' = y - y2, y(0) = 0.5; y(0.5)

Use a numerical solver and Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05.

y' = xy2 – y/x, y(1) = 1; y(1.5)

Use a numerical solver and Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05.

y' = (x - y)2, y(0) = 0.5; y(0.5)

Use a numerical solver and Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05.

y' = xy + √y, y(0) = 1; y(0.5)

Use a numerical solver and Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05.

y' = x2 + y2, y(0) = 1; y(0.5)

Use a numerical solver and Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05.

y' = e-y, y(0) = 0; y(0.5)

Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05. Find an explicit solution for each initial-value problem and then construct tables similar to tables (a) and (b).

y' = y, y(0) = 1; y(1.0)


(a)


(b)

Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05. Find an explicit solution for each initial-value problem and then construct tables similar to tables (a) and (b).

y' = 2xy, y(1) = 1; y(1.5)

(a)


(b)

Use Euler’s method to obtain a four-decimal approximation of the indicated value. Carry out the recursion of (3) by hand, fi­rst using h = 0.1 and then using h = 0.05.

y' = x + y2, y(0) = 0; y(0.2)

Use Euler’s method to obtain a four-decimal approximation of the indicated value. Carry out the recursion of (3) by hand, fi­rst using h = 0.1 and then using h = 0.05.

y' = 2x - 3y + 1, y(1) = 5; y(1.2)

In the study of population dynamics one of the most famous models for a growing but bounded population is the logistic equation

dP/dt = P(a - bP),

where a and b are positive constants. Although we will come back to this equation and solve it by an alternative method in Section 3.2, solve the DE this ­first time using the fact that it is a Bernoulli equation.


We saw that a mathematical model for the velocity v of a chain slipping off the edge of a high horizontal platform is

xv dv/dx + v2 = 32x.

In that problem you were asked to solve the DE by converting it into an exact equation using an integrating factor. This time solve the DE using the fact that it is a Bernoulli equation.

Write the differential equation in the form x(y′/y) = ln x + ln y and let u = ln y.

Then

du/dx = y′/y and the differential equation becomes x(du/dx) = ln x + u or du/dx − u/x = (ln x)/x, which is first-order and linear. An integrating factor is e−∫ dx/x = 1/x, so that (using integration by parts)

d/dx [1/x u] = ln x/x2 and u/x = −1/x – (ln x/x) + c.

The solution is

ln y = −1 − ln x + cx or y = ecx−1/x.

Determine an appropriate substitution to solve

xy' = y ln(xy).

The differential equation dy/dx = P(x) + Q(x)y + R(x)y2 is known as Riccati’s equation.

(a) A Riccati equation can be solved by a succession of two substitutions provided that we know a particular solution y1 of the equation. Show that the substitution y = y1 + u reduces Riccati’s equation to a Bernoulli equation (4) with n = 2. The Bernoulli equation can then be reduced to a linear equation by the substitution w = u-1.

(b) Find a one-parameter family of solutions for the differential equation

dy/dx = -4/x2 – 1/x y + y2

where y1 = 2yx is a known solution of the equation.

In Example 3 the solution y(x) becomes unbounded as x­: 6→ ±∞. Nevertheless, y(x) is asymptotic to a curve as x­: → -∞ and to a different curve as x → ∞. What are the equations of these curves?

(a) Determine two singular solutions of the DE in Problem 10.

(b) If the initial condition y(5) = 0 is as prescribed in Problem 10, then what is the largest interval I over which the solution is defi­ned? Use a graphing utility to graph the solution curve for the IVP.

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