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mathematics
first course differential equations

A First Course in Differential Equations with Modeling Applications 11th edition Dennis G. Zill - Solutions

A one-parameter family of solutions of the first-order differential equation dy/dx = xy1/2 is y = (1/4 x4 + c)2 for c ≥ 0. Each solution in this family is dened on (-∞, ∞). The last statement is not true if we choose c to be negative. For c = -1, explain why y = (1/4x4 – 1)2 is not a
(a) Explain why the interval of denition of the explicit solution y = φ2(x) of the initial-value problem in Example 2 is the open interval (25, 5).(b) Can any solution of the differential equation‑cross the x-axis? Do you think that x2 + y2 = 1 is an implicit solution of the initial-value
Use a technique of integration or a substitution to nd an explicit solution of the given differential equation or initialvalue problem.1. dy/dx = 1 / (1 + sinx)2. dy/dx = sin √x / √y3. (√x + x)dy/dx = √y + y4. dy/dx = y2/3 –y5. dy/dx = e√x/y, y(1) = 46. dy/dx = x tan-1 x/y, y(0) = 3
(a) The autonomous first-order differential equation dy/dx = 1/(y - 3) has no critical points. Nevertheless, place 3‑on the phase line and obtain a phase portrait of the equation. Compute d2y/dx2 to determine where solution curves are concave up and where they are concave down. Use the phase
Every autonomous first-order equation dy/dx = f (y) is separable. Find explicit solutions y1(x), y2(x), y3(x), and y4(x) of the differential equation dy/dx = y - y3 that satisfy, in turn, the initial conditions y1(0) = 2, y2(0) = ½, y3(0) = -1 and y4(0) = -2. Use a graphing utility to plot the
Solve the given differential equation by using an appropriate substitution.dy/dx = tan2(x + y)
Solve the given differential equation by using an appropriate substitution.dy/dx = sin(x + y)
Solve the given differential equation by using an appropriate substitution.dy/dx = 1 + ey-x+5
Solve the given differential equation by using an appropriate substitution.dy/dx = 2 + √(y + 2x + 3)
Solve the given initial-value problem.dy/dx = cos (x + y), y(0) = π/4
Solve the given initial-value problem.dy/dx = (3x + 2y)/(3x + 2y + 2) , y(-1) = -1
Explain why it is always possible to express any homogeneous differential equation M(x, y) dx + N(x, y) dy = 0 in the formdy/dx = F(y/x).You might start by proving thatM(x, y) = xαM(1, y/x) and N(x, y) = xαN(1, y/x).
Put the homogeneous differential equation(5x2 - 2y2) dx - xy dy = 0into the form given in Problem 31.
(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.
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?
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
Determine an appropriate substitution to solvexy' = y ln(xy).
Write the differential equation in the form x(y′/y) = ln x + ln y and let u = ln y.Thendu/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
We saw that a mathematical model for the velocity v of a chain slipping off the edge of a high horizontal platform isxv 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
In the study of population dynamics one of the most famous models for a growing but bounded population is the logistic equationdP/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
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)
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)
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) Actual value Abs. error
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) yn Actual value Abs. error % Rel.
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)
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' = 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' = (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' = 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' = y - y2, y(0) = 0.5; y(0.5)
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,
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,
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.
(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
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.
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' + k1y = k2, where k1 and k2 are nonzero constants, always possesses a constant solution.________
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.An example of a nonlinear third-order differential equation in normal form is _____
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.Every autonomous DE dy/dx = f (y) is separable.________
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.If y' = exy, then y = __________.
If a differentiable function y(x) satisfies y' = |x|, y(-1) = 2, then y(x) = ________________.
Fill in the blanks or answer true or false.is a solution of the linear first order differential equation ___________. y = ecos x te-cos s t dt
Construct an autonomous first order differential equation dy/dx = f (y) whose phase portrait is consistent with the given figure. y 3
Construct an autonomous Âfirst order differential equation dy/dx = f (y) whose phase portrait is consistent with the given Âfigure. 4
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.
Solve the given differential equation.(y2 + 1 dx) = y sec2 xdy
Solve the given differential equation.y(ln x - ln y) dx = (x ln x - x ln y - y) dy
Solve the given differential equation.(6x + 1)y2 dy/dx + 3x2 + 2y3 = 0
Solve the given differential equation.dx/dy = - 4y2 + 6xy/3y2 + 2x
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†’ˆž P(t). P
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. %3B %3B %3B
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 -
Solve the given differential equation.t dQ/dt + Q = t4 ln t
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.(2x + y + 1)y' = 1
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.(x2 + 4) dy = (2x - 8xy) dx
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
(a) Without solving, explain why the initial-value problem dy/dx = √y, y(x0) = y0 has 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.
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
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.dy/dx + P(x)y = ex, y(0) = -1, where 1, 0
Solve the given initial-value problem.dy/dx + y = f (x), y(0) = 5, where Je, 0
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
(a) Find an implicit solution of the initial-value problemdy/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.
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.
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,
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,
Make up a differential equation that does not possess any real solutions.
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 (-âˆž, âˆž).1.2. dx = x + 3y dt dy 5x + 3y, dt x = e-2 + 3e6, –21 y = -e-2 + Se6r d?x = 4y + e' dt? d²y = 4x – e'; dt? x = cos 2t + sin 2t + e', y
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' = 02. x2y'' - 7xy' + 15y = 0
In Example 5 we saw that y = Î¦1 (x) = ˆš(25 €“ x2) and y =2(x) = -ˆš(25 €“ x2) are solutions of dy/dx = -x/y on the interval (-5, 5). Explain why the peicewise-defined function IV25 - х, —5
Verify that the piecewise-defined functiois a solution of the differential equation xy'- 2y = 0 on (-ˆž, ˆž). — x, х 0
Verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution1. 2.3.4. dP dt Cje 1 + cje = P(1 – P); P = %3D dy dx + 2xy = 1; y = e¬x²| e* dt + cje-x² my
Verify that the indicated expression is an implicit solution of the given first-order differential equation. Find at least one explicit solution y Î¦ (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
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
Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution1. 2y' + y = 0; y = e-x/22. dy/dt + 20y = 24; y – 6/5 – 6/5 e-20t3. y'' – 6y' + 13y = 0; y = e3x cos 2x4. y'' + y = tan x; y = -(cos
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 x2. u dv + (v + uv ueu) du 0; in v; in u
State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6).1. $dy d³y + y = 0 \dx) х dx$
Find a singular solution of Problem 21. Of Problem 22.Data from problem 21 & 2221. dy/dx = x√(1 – y2)22. (ex + e-x)dy/dx = y2
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)
(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
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) = -12. (2y - 2)dy/dx 2x2 + 4x + 2, y(1) = -23. ey dx – e-x dy = 0, y(0) = 04.
Proceed as in Example 5 and find an explicit solution of the given initial-value problem.1. dy/dx = ye-x2, y(4) = 12. dy/dx = y2 sinx2, y(-2) = 1/3
Find an explicit solution of the given initial-value problem.1. dx/dt = 4(x2 + 2), x(π/4) =12. dy/dx = (y2 - 2)/ (x2 - 1), y(2) = 23. x2 dy/dx = y – xy, y(-1) = -14. dy/dt + 2y = 1, y(0) = 5/25. √(1 – y2) dx – √(1 – x2) dy = 0, y(0) = √3 / 26. √(1 – y4) dy + x(1 + 4y2) dx = 0,
Solve the given differential equation by separation of variables.1. dy/dx = sinx2. dy/dx = (x + 1)23. dx + e2x dy = 04. dy – (y - 1)2 dx = 05. x dy/dx = 4y6. dy/dx + 2xy2 = 07. dy/dx = e3x+2y8. ex ydy/dx = e-y + e-2x-y9. y ln xdy/dx = (y + 1 / x)210. dy/dx = (2y + 3 / 4x + 5)211. csc ydx +
Chemical Reactions When certain kinds of chemicals are combined, the rate at which the new compound is formed is modeled by the autonomous differential equationdX/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
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 17For a first-order DE dy/dx = f (x, y)
Terminal Velocity In Section 1.3 we saw that the autonomous differential equationM 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
Population Model Another population model is given bydP/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?
Population Model The differential equation in Example 3 is a well-known population model. Suppose the DE is changed todP/dt = P(aP - b),where a and b are positive constants. Discuss what happens to the population P as time t increases.
Suppose the autonomous DE in (2) has no critical points. Discuss the behavior of the solutions.
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 -
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 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 †’ ˆž y(x) = c2. Discuss why there
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

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