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mathematics
advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

What are eigenvalues? What role did they play in this chapter?
Find a real general solution of the following systems. Show the details.y'1 = y2y'2 = -y1 + y3y'3 = -y'2
Determine the type and stability of the critical point. Then find a real general solution and sketch or graph some of the trajectories in the phase plane. Show the details of your work.y'1 = y1 +
Find the location and type of all critical points by linearization. Show the details of your work.y1' = -y1 + y2 - y22y2' = -y1 - y2
Find a general solution. Show the details of your work.y'1 = -3y1 - 4y2 + 11t + 15y'2 = 5y1 + 6y2 + 3e-t - 15t - 20
What does stability mean in general? In connection with critical points? Why is stability important in engineering?
Find out experimentally how general you must choose y(p), in particular when the components of g have a different form (e.g., as in Prob. 7). Write a short report, covering also the situation in the
What does linearization of a system mean?
Find a real general solution of the following systems. Show the details.y'1 = 10y1 - 10y2 - 4y3y'2 = -10y1 + y2 - 14y3y'3 = -4y1 - 14y2 - 2y3
Determine the type and stability of the critical point. Then find a real general solution and sketch or graph some of the trajectories in the phase plane. Show the details of your work.y'1 = 4y1 +
Find the location and type of all critical points by first converting the ODE to a system and then linearizing it.y" - 9y + y3 = 0
Review the pendulum equations and their linearizations.
Solve, showing details:y'1 = y2 + 6e2ty'2 = y1 - e2ty1(0) = 1, y2(0) = 0
Find a general solution. Determine the kind and stability of the critical point.y'1 = 2y2y'2 = 8y1
Solve the following initial value problems.y'1 = 2y1 + 5y2Y'2 = -1/2y1 -3/2y2y1(0) = -12, y2(0) = 0
Solve y" + 2y' + 2y = 0. What kind of curves are the trajectories?
Find the location and type of all critical points by first converting the ODE to a system and then linearizing it.y" + cos y = 0
Find a general solution of the given ODE (a) by first converting it to a system, (b), as given. Show the details of your work.4y" - 15y' - 4y = 0
Solve, showing details:y'1 = y2 - 5 sin ty'2 = -4y1 + 17 cos ty1(0) = 5, y2(0) = 2
Find a general solution. Determine the kind and stability of the critical point.y'1 = -2y1 + 5y2y'2 = -y1 - 6y2
Solve the following initial value problems.y'1 = y2y'2 = y1y1(0) = 0, y2(0) = 2
Discuss the critical points in (10)?(13) of Sec. 4.3 by using Tables 4.1 and 4.2. Table 4.1 Eigenvalue Criteria for Critical Points (Derivation after Table 4.2) Name (a) Node (b) Saddle point (c)
Find the location and type of all critical points by first converting the ODE to a system and then linearizing it.y" + sin y = 0
Find a general solution of the given ODE (a) by first converting it to a system, (b), as given. Show the details of your work.y" + 2y' - 24y = 0
(a) In Example 2 choose a sequence of values of C that increases beyond bound, and compare the corresponding sequences of eigenvalues of A. What limits of these sequences do your numeric values
Solve, showing details:y'1 = y1 + 2y2 + e2t - 2ty'2 = -y2 + 1 + ty1(0) = 1, y2(0) = -4
Find a general solution. Determine the kind and stability of the critical point.y'1 = -3y1 - 2y2y'2 = -2y1 - 3y2
Solve the following initial value problems.y'1 = 3y1 + 2y2y'2 = 2y1 + 3y2y1(0) = 0.5, y2(0) = -0.5
What happens in Example 4 of Sec. 4.3 if you change A to A + 0.1I, where I is the unit matrix?
Write the y" - 4y + y3 = 0 as a system, solve it for y2 as a function of y1, and sketch or graph some of the trajectories in the phase plane. -2 C = 3 C = 4 ON 2 C=5 Fig. 98. Trajectories in Problem
Write a short report in which you compare the application of the method of undetermined coefficients to a single ODE and to a system of ODEs, using ODEs and systems of your choice.
Find the currents in Fig. 99 (Probs. 17?19) for the following data, showing the details of your work. R1 = 2 ?, R2 = 8 ?, L = 1 H, C = 0.5 F, E = 200 V E L m IV R₁ Switch Fig. 99. Problems 17-19
Find a general solution. Determine the kind and stability of the critical point.y'1 = -y1 + 2y2y'2 = -2y1 - y2
Show that F (y, y', y'') = 0 can be reduced to a first-order ODE with y as the independent variable and y'' = (dz/dy)z, where z = y'; derive this by the chain rule. Give two examples.
Find the Wronskian. Show linear independence by using quotients and confirm it by Theorem 2.e4.0x, e-1.5x
Find a general solution. Show the details of your calculation.(4D2 - 12D + 9I)y = 2e1.5x
Find a general solution. Show the details of your calculation.(x2D2 + 2xD – 12I)y = 0
Find a general solution. Show the details of your calculation.(2D2 - 3D - 2I)y = 13 - 2x2
What are the conditions for an RLC-circuit to be (I) overdamped, (II) critically damped, (III) underdamped? What is the critical resistance Rcrit (the analog of the critical damping constant
(a) Verify that the given functions are linearly independent and form a basis of solutions of the given ODE. (b) Solve the IVP. Graph or sketch the solution.4y'' + 25y = 0y(0) = 3.0y' (0) =
Solve and graph the solution. Show the details of your work.x2y" + 3xy' + y = 0, y(1) = 3.6, y' (1) = 0.4
Solve the initial value problem. State which rule you are using. Show each step of your calculation in detail.(x2D2 - 3xD + 3I)y = 3 ln x - 4, y(1) = 0, y' (1) = 1; yp = ln x
(a) Find a second-order homogeneous linear ODE for which the given functions are solutions. (b) Show linear independence by the Wronskian. (c) Solve the initial value problem.cosh 1.8x,
Find the transient motion of the mass–spring system modeled by the ODE. Show the details of your work.(D2 + 4D + 8I)y = 2 cos 2t + sin 2t
Solve the initial value problem for the RLC circuit in Fig. 61 with the given data, assuming zero initial current and charge. Graph or sketch the solution. Show the details of your work. R = 6 ?, L =
(a) Verify that the given functions are linearly independent and form a basis of solutions of the given ODE. (b) Solve the IVP. Graph or sketch the solution.4x2y'' - 3y = 0y(1) = -3y' (1) =
(a) Verify that the given functions are linearly independent and form a basis of solutions of the given ODE. (b) Solve the IVP. Graph or sketch the solution.y'' + 2y' + 2y = 0y(0) = 0y' (0) =
Solve and graph the solution. Show the details of your work.(x2D2 - xD - 15I)y = 0, y(1) = 0.1, y' (1) = -4.5
Find an ODE y" + ay' + by = 0 for the given basis.e(-2 +i)x, e(-2-i)x
Find the motion of the mass–spring system modeled by the ODE and the initial conditions. Sketch or graph the solution curve. In addition, sketch or graph the curve of y - yp to see when the
Solve the problem, showing the details of your work. Sketch or graph the solution.(x2D2 + xD - I)y = 16x3, y(1) = - 1, y' (1) = 1
Solve the IVP. Check that your answer satisfies the ODE as well as the initial conditions. Show the details of your work.y" + 25y = 0y(0) = 4.6y' (0) = -1.2
Derive the formula after (12) from (12). Can we have beats in a damped system?
Find the steady-state current in the RLC-circuit in Fig. 71 when R = 2 k? (2000 ?), L = 1 H, C = 4 ? 10-3 F, and E = 110 sin 415t V (66 cycles/sec). R C rele L E(t) Fig. 71. RLC-circuit
Solve the IVP. Check that your answer satisfies the ODE as well as the initial conditions. Show the details of your work.y" + y' - 6y = 0y(0) = 10y' (0) = 0
(a) Derive, in detail, the crucial formula (16). (b) By considering dC*/dc show that C* (?max) increase as c ( ?2mk) decreases. (c) Illustrate practical resonance with an ODE of your own in which you
Find the steady-state current in the RLC-circuit in Fig. 71 when R = 50 ?, L = 30 H, C = 0.025 F, E = 200 sin 4t V. R C rele L E(t) Fig. 71. RLC-circuit
Solve the IVP. Check that your answer satisfies the ODE as well as the initial conditions. Show the details of your work.The ODE in Prob. 15y" + 0.54y' + (0.0729 + π)y = 0y(0) = 0y' (0) = 1
Solve the IVP. Check that your answer satisfies the ODE as well as the initial conditions. Show the details of your work.y" - y = 0y(0) = 2y' (0) = -2
Solve the IVP. Check that your answer satisfies the ODE as well as the initial conditions. Show the details of your work.The ODE in Prob. 5y" + 2πy' + π2y = 0y(0) = 4.5y' (0) = -4.5π - 1 = 13.137
Find an electrical analog of the mass–spring system with mass 4 kg, spring constant 10 Kg/sec2, damping constant 20 kg/sec, and driving force 100 sin 4t nt.
(a) Solve the initial value problem y" + y = cos ?t, ?2 ? 1, y(0) = 0, y' (0) = 0. Show that the solution can be written (b) Experiment with the solution by changing ? to see the change of the
Show that the system in Fig. 72 with m = 4, c = 0, k = 36, and driving force 61 cos 3.1t exhibits beats. Choose zero initial conditions. m Spring Mass Dashpot Fig. 72. Mass-spring system
Linear Independence is of basic importance, in this chapter, in connection with general solutions, as explained in the text. Are the following functions linearly independent on the given interval?
Solve and graph the solution. Show the details of your work.(x2D2 + xD + I)y = 0, y(1) = 1, y' (1) = 1
Solve the initial value problem. State which rule you are using. Show each step of your calculation in detail.(D2 + 0.2D + 0.26I)y = 1.22e0.5x, y(0) = 3.5, y' (0) = 0.35
Find an ODE y" + ay' + by = 0 for the given basis.e-√5xxe-√5x
Find the motion of the mass–spring system modeled by the ODE and the initial conditions. Sketch or graph the solution curve. In addition, sketch or graph the curve of y - yp to see when the
Determine the values of t corresponding to the maxima and minima of the oscillation y(t) = e-t sin t. Check your result by graphing y(t).
Explain Table 2.2 in a 1?2 page report with examples, e.g., the analog (with L = 1 H) of a mass?spring system of mass 5 kg, damping constant 10 kg/sec, spring constant 60 kg/sec2, and driving force
Solve the problem, showing the details of your work. Sketch or graph the solution.y" + 16y = 17ex, y(0) = 6, y' (0) = -2
Find a general solution. Show the details of your calculation.(100D2 - 160D + 64I)y = 0
Find the steady-state current in the RLC-circuit in Fig. 61 for the given data. Show the details of your work. R = 12, L = 1.2 H, C = 20/3 ? 10-3?F, E = 12,000 sin 25t V R с E(t) = Esin cot
If, in the motion of a small body on a straight line, the sum of velocity and acceleration equals a positive constant, how will the distance y(t) depend on the initial velocity and position?
Solve and graph the solution. Show the details of your work.x2y" + 3xy' + 0.75y = 0, y(1) = 1, y' (1) = -1.5
Find the critical motion (8) that starts from y0 with initial velocity v0. Graph solution curves for α = 1, y0 = 1 and several v0 such that (i) the curve does not intersect the t-axis, (ii) it
(a) Find a second-order homogeneous linear ODE for which the given functions are solutions. (b) Show linear independence by the Wronskian. (c) Solve the initial value problem.1 , e-2x, y(0)
Find the transient motion of the mass–spring system modeled by the ODE. Show the details of your work.(D2+ I)y = cos ωt, ω2 ≠ 1
Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your work.(x2D2 + xD - 9I)y = 48x5
Illustrate the linearity of L in (2) by taking c = 4, k = -6, y = e2x, and w = cos 2x. Prove that L is linear.
Find a general solution. Show the details of your calculation.y" + 6y' + 34y = 0
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
(a) Find a second-order homogeneous linear ODE for which the given functions are solutions.(b) Show linear independence by the Wronskian.(c) Solve the initial value problem.cos 5x, sin 5x, y(0) = 3,
Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your work.(D2 - 2D + l)y = 35x3/2 ex
Find a (real) general solution. State which rule you are using. Show each step of your work.(D2 – 16I)y = 9.6e4x + 30ex
Apply the given operator to the given functions. Show all steps in detail.(D2 - 4.20D + 4.4I/)y = 0
Find a real general solution. Show the details of your work.(x2D2 - 3xD + 10I)y = 0
Find a real general solution. Show the details of your work.(x2D2 - 0.2xD + 0.36I)y = 0
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 amplitude of the steady-state current (5) becomes
Reduce to first order and solve, showing each step in detail.y'' + y'3 sin y = 0
Find a general solution. Show the details of your calculation.4y" + 32y' + 63y = 0
Find the frequency of oscillation of a pendulum of length L (Fig. 42), neglecting air resistance and the weight of the rod, and assuming ? to be so small that sin ? practically equals ?. L. Body of
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
Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your work.(D2 - 4D + 4I)y = 6e2x/x4
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
Apply the given operator to the given functions. Show all steps in detail.(4D2 - l)y = 0
Find a real general solution. Show the details of your work.(x2D2 - 4xD + 6I)y = C
What do you know about existence and uniqueness of solutions of linear second-order ODEs?
What is resonance? How can you remove undesirable resonance of a construction, such as a bridge, a ship, or a machine?
Describe applications of ODEs in mechanical systems. What are the electrical analogs of the latter?

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