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Q: Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the
Q: Graph a direction field (by a CAS or by hand). In the field graph several solution curves by hand, particularly those passing through the given
Q: What is an exact ODE? Is f(x) dx + g(y) dy = 0 always exact?
Q: Explain the idea of an integrating factor. Give two examples.
Q: Test for exactness. If exact, solve. If not, use an integrating factor as given or obtained by inspection or by the theorems in the text. Also, if an
Q: Find a general solution. Show the steps of derivation. Check your answer by substitution.xy' = y + 2x3 sin2 y/x (Set y/x = u)
Q: Solve the ODE by integration or by remembering a differentiation formula.y' = cosh 5.13x
Q: Sketch or graph some of the given curves. Guess what their OTs may look like. Find these OTs.y = c/x2
Q: Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the
Q: Graph a direction field (by a CAS or by hand). In the field graph several solution curves by hand, particularly those passing through the given
Q: Can an ODE sometimes be solved by several methods? Give three examples.
Q: What does modeling mean? Can a CAS solve a model given by a first-order ODE? Can a CAS set up a model?
Q: Give problems from mechanics, heat conduction, and population dynamics that can be modeled by first-order ODEs.
Q: Test for exactness. If exact, solve. If not, use an integrating factor as given or obtained by inspection or by the theorems in the text. Also, if an
Q: Find a general solution. Show the steps of derivation. Check your answer by substitution.xy' = y2 + y (Set y/x = u)
Q: Can two solution curves of the same ODE have a common point in a rectangle in which the assumptions of the present theorems are satisfied?
Q: (a) Verify that y is a solution of the ODE.(b) Determine from y the particular solution of the IVP(c) Graph the solution of the IVP.y' + 4y = 1.4y =
Q: Sketch or graph some of the given curves. Guess what their OTs may look like. Find these OTs.y = ce-x2
Q: Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the
Q: Direction fields are very useful because they can give you an impression of all solutions without solving the ODE, which may be difficult or even
Q: What is the largest possible α in Example 1 in the text?Data from example 1Consider the initial value problemand take the rectangle R; |x| < 5,
Q: Graph a direction field (by a CAS or by hand) and sketch some solution curves. Solve the ODE exactly and compare.y' + 2y = 0
Q: Test for exactness. If exact, solve. If not, use an integrating factor as given or obtained by inspection or by the theorems in the text. Also, if an
Q: Solve the IVP. Show the steps of derivation, beginning with the general solution.xy' + y = 0 y(4) = 6
Q: (a) Verify that y is a solution of the ODE.(b) Determine from y the particular solution of the IVP(c) Graph the solution of the IVP.y' = y + exy = (x
Q: Let the electric equipotential lines (curves of constant potential) between two concentric cylinders with the z-axis in space be given by u(x, y) =
Q: Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the
Q: This means an ODE not showing x (the independent variable) explicitly. What will the level curves f (x, y) = const (also called isoclines = curves of
Q: Graph a direction field (by a CAS or by hand) and sketch some solution curves. Solve the ODE exactly and compare.y' = y - 4y2
Q: Test for exactness. If exact, solve. If not, use an integrating factor as given or obtained by inspection or by the theorems in the text. Also, if an
Q: Solve the IVP. Show the steps of derivation, beginning with the general solution.y' cosh2 x = sin2 yy(0) = 1/2π
Q: (a) Verify that y is a solution of the ODE.(b) Determine from y the particular solution of the IVP(c) Graph the solution of the IVP.y' = y - y2y =
Q: Let the isotherms (curves of constant temperature) in a body in the upper half-plane y > 0 be given by 4x2 + 9y2 = c. Find the orthogonal
Q: Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the
Q: Model the motion of a body B on a straight line with velocity as given, y(t) being the distance of B from a point y = 0 at time t. Graph a direction
Q: Graph a direction field (by a CAS or by hand) and sketch some solution curves. Solve the ODE exactly and compare.y' + y = 1.01 cos 10x
Q: Under what conditions for the constants a, b, k, l is (ax + by) dx + (kx + ly) dy = 0 exact? Solve the exact ODE.
Q: Solve the IVP. Show the steps of derivation, beginning with the general solution.y' = -4x/yy(2) = 3
Q: Show that for a family u(x, y) = c = const the orthogonal trajectories v(x, y) = c* = const can be obtained from the following Cauchy–Riemann
Q: Find the general solution. Indicate which method in this chapter you are using. Show the details of your work.y' + 2.5y = 1.6x
Q: Working backward from the solution to the problem is useful in many areas. Euler, Lagrange, and other great masters did it. To get additional insight
Q: Solve the IVP. Show the steps of derivation, beginning with the general solution.xy' = y + 3x4 cos2 (y/x)y(1) = 0(Set y/x = u)
Q: This is the simplest method to explain numerically solving an ODE, more precisely, an initial value problem (IVP). Using the method, to get a feel
Q: Find the general solution. Indicate which method in this chapter you are using. Show the details of your work.25yy' - 4x = 0
Q: If the growth rate of the number of bacteria at any time t is proportional to the number present at t and doubles in 1 week, how many bacteria can be
Q: In dropping a stone or an iron ball, air resistance is practically negligible. Experiments show that the acceleration of the motion is constant
Q: Find the general solution. Indicate which method in this chapter you are using. Show the details of your work.(3xey + 2y) dx + (x2ey + x)
Q: What should be the 146C content (in percent of y0) of a fossilized tree that is claimed to be 3000 years old?
Q: Another method of obtaining (4) results from the following idea. Write (3) as cy*where y* is the exponential function, which is a solution of
Q: Solve the IVP. Indicate the method used. Show the details of your work.y' = √1 - y2y(0) = 1/√2
Q: Experiments show for a gas at low pressure p (and constant temperature) the rate of change of the volume V(p) equals -V/p. Solve the model.
Q: Using a method of this section or separating variables, find the general solution. If an initial condition is given, find also the particular
Q: Solve the IVP. Indicate the method used. Show the details of your work.3 sec y dx + 1/3 sec x dy = 0, y(0) = 0
Q: If the growth rate of a culture of bacteria is proportional to the number of bacteria present and after 1 day is 1.25 times the original number,
Q: Using a method of this section or separating variables, find the general solution. If an initial condition is given, find also the particular
Q: If in a reactor, uranium 23797 U loses 10% of its weight within one day, what is its half-life? How long would it take for 99% of the original amount
Q: We have transformed ODEs to separable form, to exact form, and to linear form. The purpose of such transformations is an extension of solution
Q: If the temperature of a cake is 300°F when it leaves the oven and is 200°F ten minutes later, when will it be practically equal to the room
Q: Find and solve the model for drug injection into the bloodstream if, beginning at t = 0, a constant amount A g/min is injected and the drug is
Q: A CAS can usually graph solutions, even if they are integrals that cannot be evaluated by the usual analytical methods of calculus.(a) Show this for
Q: Lake Erie has a water volume of about 450 km3 and a flow rate (in and out) of about 175 km2 per year. If at some instant the lake has pollution
Q: In Prob. 36 find and graph the solution satisfying y(0) = 2 when (for simplicity) A = B = 1 and H = 0.2. What is the limit? What does it mean? What
Q: If in a population y(t) the death rate is proportional to the population, and the birth rate is proportional to the chance encounters of meeting
Q: Verify directly by substitution that x(1-a)/2 ln x is a solution of (1) if (2) has a double root, but xm1 and xm2 ln x are not
Q: Apply the given operator to the given functions. Show all steps in detail.D2 + 2D; cosh 2x, e-x + e2x, cos x
Q: Find a (real) general solution. State which rule you are using. Show each step of your work.y" + 5y' + 4y = 10e-3x
Q: Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your work.y" + 9y = sec 3x
Q: Find a general solution. Check your answer by substitution. ODEs of this kind have important applications to be discussed in Secs. 2.4, 2.7, and
Q: Find the harmonic motion (4) that starts from y0 with initial velocity v0. Graph or sketch the solutions for ω0 = π, y0 = 1, and various
Q: Why are linear ODEs preferable to nonlinear ones in modeling? What does an initial value problem of a second-order ODE look like? Why must you have a
Q: Show that F (x, y', y'') = 0 can be reduced to first order in z = y' (from which y follows by integration). Give two examples of your own.
Q: What does an initial value problem of a second-order ODE look like? Why must you have a general solution to solve it?
Q: Find a real general solution. Show the details of your work.5x2y" + 23xy' + 16.2y = 0
Q: Apply the given operator to the given functions. Show all steps in detail.(D - 2I)2; e2x, xe2x, e-2x
Q: Find a (real) general solution. State which rule you are using. Show each step of your work.y" + 3y' + 2y = 12x2
Q: Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your
Q: Find the steady-state motion of the mass–spring system modeled by the ODE. Show the details of your work.y" + 6y' + 8y = 42.5 cos 2t
Q: How does the frequency of the harmonic oscillation change if we (i) double the mass, (ii) take a spring of twice the modulus? First find qualitative
Q: By what methods can you get a general solution of a nonhomogeneous ODE from a general solution of a homogeneous one?
Q: Reduce to first order and solve, showing each step in detail.y'' + y' = 0
Q: Describe applications of ODEs in mechanical systems. What are the electrical analogs of the latter?
Q: What is resonance? How can you remove undesirable resonance of a construction, such as a bridge, a ship, or a machine?
Q: What do you know about existence and uniqueness of solutions of linear second-order ODEs?
Q: Find a real general solution. Show the details of your work.(x2D2 - 4xD + 6I)y = C
Q: Apply the given operator to the given functions. Show all steps in detail.(4D2 - l)y = 0
Q: Find a (real) general solution. State which rule you are using. Show each step of your work.(D2 + 2D + 3/4l)y = 3ex + 9/2x
Q: Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your
Q: Find the steady-state motion of the mass–spring system modeled by the ODE. Show the details of your work.(4D2 + 12D + 9I)y = 225 - 75 sin 3t
Q: Find the frequency of oscillation of a pendulum of length L (Fig. 42), neglecting air resistance and the weight of the rod, and assuming θ to be so
Q: Find a general solution. Show the details of your calculation.4y" + 32y' + 63y = 0
Q: Reduce to first order and solve, showing each step in detail.y'' + y'3 sin y = 0
Q: In tuning a stereo system to a radio station, we adjust the tuning control (turn a knob) that changes C (or perhaps L) in an RLC-circuit so that the
Q: Find a real general solution. Show the details of your work.(x2D2 - 0.2xD + 0.36I)y = 0
Q: Find a real general solution. Show the details of your work.(x2D2 - 3xD + 10I)y = 0
Q: Apply the given operator to the given functions. Show all steps in detail.(D2 - 4.20D + 4.4I/)y = 0
Q: Find a (real) general solution. State which rule you are using. Show each step of your work.(D2 – 16I)y = 9.6e4x + 30ex
Q: Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your
Q: (a) Find a second-order homogeneous linear ODE for which the given functions are solutions.(b) Show linear independence by the Wronskian.(c) Solve
Q: Find the transient motion of the mass–spring system modeled by the ODE. Show the details of your work.y" + 3y' + 3.25y = 3 cos t - 1.5 sin t
Q: Find a general solution. Show the details of your calculation.y" + 6y' + 34y = 0
Showing 1 - 100 of 1703
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