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advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Find the maximum flow by Ford-Fulkerson: 4, 2 (1) S 3,2 (2) 2, 1 (3) 5, 3 3, 2 6,3 (4) 10, 4 3, 1 5) 1,0 (6)
This is just a sample of such ODEs; some more follow in the next problem set. Find a general solution in terms of Jv and J-v or indicate when this is not possible. Use the indicated
Find a general solution in terms of Jv and Yv. Indicate whether you could also use J-v instead of Yv. Use the indicated substitution. Show the details of your work.y" + k2x2y = 0 (y = u√x, 1/2kx2 =
Apply the power series method. Do this by hand, not by a CAS, to get a feel for the method, e.g., why a series may terminate, or has even powers only, etc. Show the details.y' = -2xy
From what n on will your CAS no longer produce faithful graphs of Pn(x)? Why?
What is the hypergeometric equation? Where does the name come from?
This is just a sample of such ODEs; some more follow in the next problem set. Find a general solution in terms of Jv and J-v or indicate when this is not possible. Use the indicated
Can a power series solution reduce to a polynomial? When? Why is this important?
Find a basis of solutions by the Frobenius method. Try to identify the series as expansions of known functions. Show the details of your work.xy" + (2x + 1)y' + (x + 1)y = 0
This is just a sample of such ODEs; some more follow in the next problem set. Find a general solution in terms of Jv and J-v or indicate when this is not possible. Use the indicated
Find a general solution in terms of Jv and Yv. Indicate whether you could also use J-v instead of Yv. Use the indicated substitution. Show the details of your work.4xy" + 4y' + y = 0 (√x = z)
Determine the radius of convergence. Show the details of your work. 00 -m-0 2m x²
Write down the most important ODEs in this chapter from memory.
This is just a sample of such ODEs; some more follow in the next problem set. Find a general solution in terms of Jv and J-v or indicate when this is not possible. Use the indicated
Determine the radius of convergence. Show the details of your work. m-0 2m+1 x (2m + 1)!
List the three cases of the Frobenius method, and give examples of your own.
Verify that the polynomials in (11') satisfy (1).
Find a basis of solutions by the Frobenius method. Try to identify the series as expansions of known functions. Show the details of your work.xy" + 2y' + xy = 0
This is just a sample of such ODEs; some more follow in the next problem set. Find a general solution in terms of Jv and J-v or indicate when this is not possible. Use the indicated
Find a general solution in terms of Jv and Yv. Indicate whether you could also use J-v instead of Yv. Use the indicated substitution. Show the details of your work.9x2y" + 9xy' + (36x4 - 16)y = 0 (x2
What is the indicial equation? Why is it needed?
This is just a sample of such ODEs; some more follow in the next problem set. Find a general solution in terms of Jv and J-v or indicate when this is not possible. Use the indicated
Determine the radius of convergence. Show the details of your work. Σ (m + 1)mxm m-0
What is the difference between the two methods in this chapter? Why do we need two methods?
Show that (7) with n =1 gives y2(x) = P1(x) = x and (6) gives 3₁ = 1 = x²-x² 1 - x2 = 1 1 1+x 1- x x ln- 1
Show that (6) with n = 0 gives P0(x) = 1 and (7) gives (use ln (1 + x) = x - 1/2x2 + 1/3x3 + ....) Verify this by solving (1) with n = 0, setting z = y' and separating variables. 32(x)=x+ 228 +20+
Write a report of 2–3 pages explaining the difference between the two methods. No proofs. Give simple examples of your own.
Show that the series (11) converges for all x. Why is the convergence very rapid?
Find a general solution in terms of Jv and Yv. Indicate whether you could also use J-v instead of Yv. Use the indicated substitution. Show the details of your work.x2y" + xy' + (x2 - 16) y = 0
(a) Write a review (2–3 pages) on power series in calculus. Use your own formulations and examples—do not just copy from textbooks. No proofs. (b) Collect and arrange Maclaurin series in a
Why are we looking for power series solutions of ODEs?
Find the location and kind of all critical points of the given nonlinear system by linearization.y'1 = -4y2y'2 = sin y1
Find the location and kind of all critical points of the given nonlinear system by linearization.y'1 = y2y'2 = y1 - y13
Find the currents in Fig. 102 when R = 2.5 ?, L = 1 H, C = 0.04 F, E(t) = 169 sin t V, I1(0) = 0, I2(0) = 0. E 1₁ R 1₂ с L еее Fig. 102. Network in Problem 25
Find a general solution. Show the details of your work.y'1 = y1 + 4y2 - 2 cos ty'2 = y1 + y2 - cos t + sin t
Find a general solution. Show the details of your work.y'1 = 4y2y'2 = 4y1 + 32t2
Graph some of the figures in this section, in particular Fig. 87 on the degenerate node, in which the vector y(2) depends on t. In each figure highlight a trajectory that satisfies an initial
Find the currents in Fig. 100 (Prob. 20) for the following data, showing the details of your work. R1 = 1 ?, R2 = 1.4 ?, L1 = 0.8 H, L2?= 1 H, E = 100 V, I1(0) = I2(0) = 0 E L₁ m 1₂V R₁ L₂ m
Stability concepts are basic in physics and engineering. Write a two-part report of 3 pages each (A) on general applications in which stability plays a role (be as precise as you can), and (B) on
Show that a model for the currents I1(t) and I2(t) in Fig. 89 is Find a general solution, assuming that R = 3 ?, L = 4 H, C = 1/12 F. 히 [dt + R(I1 - Iz) = 0, LI2 + R(Iz - Ii) = 0. - = с 46 R www
What kind of critical point does y' = Ay have if A has the eigenvalues2 + 3i, 2 - 3i
Find the currents in Fig. 99 (Probs. 17?19) for the following data, showing the details of your work. In Prob. 17 find the particular solution when currents and charge at t = 0 are zero. Data from
Graph phase portraits for the systems in Prob. 17 with the values of b suggested in the answer. Try to illustrate how the phase portrait changes “continuously” under a continuous change of b.Data
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 = -2y1 +
How can you transform an ODE into a system of ODEs?
What is population dynamics? Give examples.
To what state (position, speed, direction of motion) do the four points of intersection of a closed trajectory with the axes in Fig. 93b correspond? The point of intersection of a wavy curve with the
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 = y2y'2 =
Find a real general solution of the following systems. Show the details.y'1 = y1 + y2y'2 = 3y1 - y2
State some applications that can be modeled by systems of ODEs.
Solve the IVP. Show the details of your work.(D3 + 9D2 + 23D + 15I)y = 12exp(-4x), y(0) = 9, Dy(0) = -41, D2y(0) = 189
Solve the IVP. Show the details of your work.y"' + 5y" + 24y' + 20y = x, y(0) = 1.94, y'(0) = -3.95, y" = -24
Starting with a basis, find third-order linear ODEs with variable coefficients for which the reduction to second order turns out to be relatively simple.
Solve the given ODE. Show the details of your work.4x3y"' + 3xy' - 3y = 10
Write a report on the method of undetermined coefficients and the method of variation of parameters, discussing and comparing the advantages and disadvantages of each method. Illustrate your findings
Solve the IVP by a CAS, giving a general solution and the particular solution and its graph.yiv + 0.45y"' - 0.165y" + 0.0045y' - 0.00175y = 0, y(0) = 17.4, y' (0) = -2.82, y" (0) = 2.0485, y"'
Solve the given ODE. Show the details of your work.(D3 + 6D2 + 12D + 8I)y = 8x2
Are the given functions linearly independent or dependent on the half-axis x > 0? Give reason.sin x, cos x, sin 2x
Solve the given IVP, showing the details of your work.(D3 - 4D)y = 10 cos x + 5 sin x, y(0) = 3, y' (0) = -2, y" (0) = -1
Solve the IVP by a CAS, giving a general solution and the particular solution and its graph.yv - 5y"' + 4y' = 0, y(0) = 3, y' (0) = -5, y" (0) = 11, y"' (0) = -23, yiv (0) = 47
Solve the IVP by a CAS, giving a general solution and the particular solution and its graph.yiv - 9y" - 400y = 0, y(0) = 0, y' (0) = 0, y" (0) = 41, y "' (0) = 0
Solve the given ODE. Show the details of your work.y"' + 4.5y" + 6.75y' + 3.375y = 0
Are the given functions linearly independent or dependent on the half-axis x > 0? Give reason.ex cos x, ex sin x, ex
Solve the given IVP, showing the details of your work.(D3 - 2D2 - 3D)y = 74e-3x sin x, y(0) = -1.4, y' (0) = 3.2, y" (0) = -5 .2
Solve the IVP by a CAS, giving a general solution and the particular solution and its graph.yiv + 4y = 0, y(0) = 1/2, y' (0) = -3/2, Y" (0) = 5/2, y"' (0) = -7/2
Solve the IVP by a CAS, giving a general solution and the particular solution and its graph.4y"' + 8y" + 41y' + 37y = 0, y(0) = 9, y' (0) = -6.5, y" (0) = -39.75
Solve the given ODE. Show the details of your work.(D4 - 16I)y = 15 cosh x
Are the given functions linearly independent or dependent on the half-axis x > 0? Give reason.tan x, cot x, 1
Solve the given IVP, showing the details of your work.yiv + 5y" + 4y = 90 sin 4x, y(0) = 1, y' (0) = 2, y" (0) = - 1, y"' (0) = -32
Solve the IVP by a CAS, giving a general solution and the particular solution and its graph.y"' + 7.5y" + 14.25y' - 9.125y = 0, y(0) = 10.05, y' (0) = -54.975, y" (0) = 257.5125
Solve the IVP by a CAS, giving a general solution and the particular solution and its graph.y"' + 3.2y" + 4.81y' = 0, y(0) = 3.4, y' (0) = -4.6, y" (0) = 9.91
These properties are important in obtaining new solutions from given ones. Therefore extend Team Project 38 in Sec. 2.2 to nth-order ODEs. Explore statements on sums and multiples of solutions of (1)
Solve the following ODEs, showing the details of your work.(D3 - 9D2 + 27D - 27I)y = 27 sin 3x
To get a feel for higher order ODEs, show that the given functions are solutions and form a basis on any interval. Use Wronskians. In Prob. 6, x > 0,1, x2, x4, x2y"' - 3xy" + 3y' = 0
Solve the given ODE. Show the details of your work.(D4 + 10D2 + 9l) y = 0
What is the Wronskian? What is it used for?
To get a feel for higher order ODEs, show that the given functions are solutions and form a basis on any interval. Use Wronskians. In Prob. 6, x > 0,1, e-x cos 2x, e-x sin 2x, y"' + 2y" + 5y' = 0
Solve the following ODEs, showing the details of your work.(x3D3 + x2D2 - 2xD + 2I)y = x-2
What form does an initial value problem for an nth order linear ODE have?
To get a feel for higher order ODEs, show that the given functions are solutions and form a basis on any interval. Use Wronskians. In Prob. 6, x > 0,e-4x, xe-4x, x2e-4x, y"' + 12y" + 48y' + 64y = 0
Solve the given ODE. Show the details of your work.yiv + 4y" = 0
If you know a general solution of a homogeneous linear ODE, what do you need to obtain from it a general solution of a corresponding nonhomogeneous linear ODE?
To get a feel for higher order ODEs, show that the given functions are solutions and form a basis on any interval. Use Wronskians. In Prob. 6, x > 0,cos x, sin x, x cos x, x sin x, yiv + 2y"
Solve the following ODEs, showing the details of your work.(D4 + 10D2 + 9I)y = 6.5 sinh 2x
List some other basic theorems that extend from second-order to nth-order ODEs.
To get a feel for higher order ODEs, show that the given functions are solutions and form a basis on any interval. Use Wronskians. In Prob. 6, x > 0,ex, e-x, e2x, y"' - 2y" - y' + 2y = 0
Solve the given ODE. Show the details of your work.y"' + 25y' = 0
What is the superposition or linearity principle? For what nth-order ODEs does it hold?
To get a feel for higher order ODEs, show that the given functions are solutions and form a basis on any interval. Use Wronskians. In Prob. 6, x > 0,1, x, x2, x3, yiv = 0
Solve the following ODEs, showing the details of your work.y"' + 3y" + 3y' + y = ex - x - 1
Solve y" - y = 0 for the initial conditions y(0) = 1, y'(0) = -1. Then change the initial conditions to y(0) = 1.001, y' (0) = -0.999 and explain why this small change of 0.001 at t = 0
Find a real general solution of the following systems. Show the details.y'1 = y1 + 2y2y'2 = 1/2y1 + y2
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 = y2y'2 =
Convert the van der Pol equation to a system. Graph the limit cycle and some approaching trajectories for μ = 0.2, 0.4, 06, 0.8, 1.0, 1.5, 2.0. Try to observe how the limit cycle changes its form
Find a general solution. Show the details of your work.y'1 = y2 + e3ty'2 = y1 - 3e3t
What are qualitative methods for systems? Why are they important?
What is the phase plane? The phase plane method? A trajectory? The phase portrait of a system of ODEs?
Find a real general solution of the following systems. Show the details.y'1 = 2y1 + 5y2y'2 = 5y1 + 12.5y2
Find the location and type of all critical points by linearization. Show the details of your work.y'1 = y2y2' = -y1 + 1/2y21
Find a general solution. Show the details of your work.y'1 = 4y1 + y2 + 0.6ty'2 = 2y1 + 3y2 - 2.5t

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