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mathematics
advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Find a (real) general solution. State which rule you are using. Show each step of your work.(D2 + 2D + I)y = 2x sin x
Find an ODE y" + ay' + by = 0 for the given basis.cos 2πx, sin 2πx
If a body hangs on a spring s1 of modulus k1 = 8, which in turn hangs on a spring s2 of modulus k2 = 12, what is the modulus k of this combination of springs?
Find the Wronskian. Show linear independence by using quotients and confirm it by Theorem 2.e-x cos ωx, e-x sin ωx
Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your work.(D2 + 6D + 9I)y = 16e-3x/(x2 + 1)
Find a (real) general solution. State which rule you are using. Show each step of your work.y" + y' + (π2 + 1/4)y = e-x/2 sin πx
Find a real general solution. Show the details of your work.x2y" + 0.7xy' - 0.1y = 0
Could you make a harmonic oscillation move faster by giving the body a greater initial push?
In Fig. 72, let m = 1 kg, c = 4 kg/sec, k = 24 kg/sec2, and r(t) = 10 cos ωt nt. Determine w such that you get the steady-state vibration of maximum possible amplitude. Determine this amplitude.
Find the motion of the mass??spring system in Fig. 72 with mass 0.125 kg, damping 0, spring constant 1.125 kg/sec2, and driving force cos t - 4 sin t nt, assuming zero initial displacement and
Find the current in the RLC-circuit in Fig. 71 when R = 40 Ω, L = 0.4 H, C = 10-4F, E = 220 sin 314t V (50 cycles/sec). R E(t) Fig. 71. RLC-circuit
Find a general solution of the homogeneous linear ODE corresponding to the ODE in Prob. 23. Data from Prob. 23 Find the steady-state current in the RLC-circuit in Fig. 71 when R = 2 kΩ (2000 Ω), L
Solve the problem, showing the details of your work. Sketch or graph the solution.(x2D2 + 15xD + 49I)y = 0, y(1) = 2, y' (1) = -11
(a) Derive a second linearly independent solution of (1) by reduction of order; but instead of using (9), Sec. 2.1, perform all steps directly for the present ODE (1).(b) Obtain xm ln x by
Solve the problem, showing the details of your work. Sketch or graph the solution.y" - 3y' + 2y = 10 sin x, y(0) = 1, y' (0) = -6
Solve LI∼" + RI∼' + I∼/C = E0eiωt, i = √-1, by substituting lp = Keiωt (K unknown) and its derivatives and taking the real part lp of the solution l∼p. Show agreement with (2),
Solve the initial value problem. State which rule you are using. Show each step of your calculation in detail.(D2 + 2D + 10I)y = 17 sin x - 37 sin 3x, y(0) = 6.6, y' (0) = -2.2
Solve and graph the solution. Show the details of your work.(9x2D2 + 3xD + I)y = 0, y(1) = 1, y' (1) = 0
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 = 18 Ω, L
Find a general solution. Show the details of your calculation.yy" = 2y'2
Show that the maxima of an under damped motion occur at equidistant t-values and find the distance.
Find an ODE y" + ay' + by = 0 for the given basis.e2.6x, e-4.3x
Solve the initial value problem. State which rule you are using. Show each step of your calculation in detail.(D2 - 2D)y = 6e2x - 4e-2x, y(0) = -1, y' (0) = 6
Solve and graph the solution. Show the details of your work.(x2D2 - 3xD + 4I)y = 0, y(1) = -π, y' (1) = 2π
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 = 8 Ω, L
Find a general solution. Show the details of your calculation.(D2 + 2D + 2I)y = 3e-x cos 2x
Find a general solution. Show the details of your calculation.(x2D2 + xD – 9I)y = 0
What is the smallest value of the damping constant of a shock absorber in the suspension of a wheel of a car (consisting of a spring and an absorber) that will provide (theoretically) an
(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.e-kx cos πx, e-kx
The undetermined-coefficient method should be used whenever possible because it is simpler. Compare it with the present method as follows.(a) Solve y" + 4y' + 3y = 65 cos 2x by both methods, showing
Solve the initial value problem. State which rule you are using. Show each step of your calculation in detail.y" + 4y' + 4y = e-2x sin 2x, y(0) = 1, y' (0) = -1.5
Solve and graph the solution. Show the details of your work.x2y" + xy' + 9y = 0, y(1) = 0, y' (1) = 2.5
Find a general solution. Show the details of your calculation.(D2 + 4πD + 4π2I)y = 0
Show that in the overdamped case, the body can pass through y = 0 at most once (Fig. 37). (b) (a) (1) Positive (2) Zero Initial velocity Negative
(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.x2, x2 lnx, y(1) =
Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your work.(D2 - I)y = 1/cosh x
Solve the initial value problem. State which rule you are using. Show each step of your calculation in detail.y" + 4y = -12 sin 2x, y(0) = 1.8, y' (0) = 5.0
Solve and graph the solution. Show the details of your work.x2y " - 4xy' + 6y = 0, y(1) = 0.4, y' (1) = 0
Find the steady-state current in the RLC-circuit in Fig. 61 for the given data. Show the details of your work. R = 0.2 Ω, L = 0.1 H, C = 2 F, E = 220 sin 314t V L E(t) = E, sin ot Fig. 61.
Find a general solution. Show the details of your calculation.y" + 0.20y' + 0.17y = 0
The unifying power of mathematical methods results to a large extent from the fact that different physical (or other) systems may have the same or very similar models. Illustrate this for the
(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.xm1, xm2, y(1) =
Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your work.(D2 + 2D + 2I)y = 4e-x sec3x
Find a real general solution. Show the details of your work.(x2D2 - xD + 5I)y = 0
Find a general solution. Show the details of your calculation.y" + y' - 12y = 0
Find the Wronskian. Show linear independence by using quotients and confirm it by Theorem 2.xk cos (In x), xk sin (In x)
Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your work.(D2 + 4I)y = cosh 2x
Find a (real) general solution. State which rule you are using. Show each step of your work.(3D2 + 27I)y = 3 cos x + cos 3x
Find a real general solution. Show the details of your work.(x2D2 - 3xD + 4I)y = 0
Find a real general solution. Show the details of your work.x2y" - 20y = 0
Find a (real) general solution. State which rule you are using. Show each step of your work.10y" + 50y' + 57.6y = cos x
Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your work.y" + 9y = csc 3x
If a weight of 20 nt (about 4.5 lb) stretches a certain spring by 2 cm, what will the frequency of the corresponding harmonic oscillation be? The period?
Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your work.y" - 4y' + 5y = e2x csc x
Find a (real) general solution. State which rule you are using. Show each step of your work.y" - 9y = 18 cos πx
Find a real general solution. Show the details of your work.xy" + 2y' = 0
Linear accelerators are used in physics for accelerating charged particles. Suppose that an alpha particle enters an accelerator and undergoes a constant acceleration that increases the speed of the
Find the steady-state motion of the mass–spring system modeled by the ODE. Show the details of your work.(D2 + 4D + 3I)y = cos t + 1/3 cos 3t
Find the steady-state motion of the mass–spring system modeled by the ODE. Show the details of your work.(D2 + D + 4.25I)y = 22.1 cos 4.5t
Solve Prob. 3 when E = E0sin ωt and R, L, E0, and?are arbitrary. Sketch a typical solution. Data from Prob. 3 Model the RL-circuit in Fig. 66. Find a general solution when R, L, E are any
Solve Prob. 1 when E = E0sin ωt and R, C, E0, and are arbitrary. Data from prob. 1 Model the RC-circuit in Fig. 64. Find the current due to a constant E. R E(t) Fig. 64. RC-circuit
Find the steady-state current in the RLC- circuit in Fig. 61 for the given data. Show the details of your work. R = 4 Ω, L = 0.5 H, C = 0.1 F, E = 500 sin 2t V L E(t) = E, sin ot Fig. 61.
Find the steady-state current in the RLC-circuit in Fig. 61 for the given data. Show the details of your work. R = 2 Ω, L = 1 H, C = 1/20 F, E = 157 sin 3t V L E(t) = E, sin ot Fig. 61. RLC-circuit
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?
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?
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 the IVP. Check that your answer satisfies the ODE as well as the initial conditions. Show the details of your work.9y" - 30y' + 25y = 0, y(0) = 3.3, y' (0) = 10.0
Solve the IVP. Check that your answer satisfies the ODE as well as the initial conditions. Show the details of your work.8y" - 2y' - y = 0, y(0) = -0.2, y' (0) = -0.325
Solve the IVP. Check that your answer satisfies the ODE as well as the initial conditions. Show the details of your work.y" - k2y = 0 (k ≠ 0), y(0) = 1, y' (0) = 1
Solve y" + y = 1 - t2/π2?if 0<t<π and 0 if t ?? ??; here, y(0) = 0, y' (0) = 0. This models an undamped system on which a force F acts during some interval of time (see Fig. 59), for
Solve the IVP. Check that your answer satisfies the ODE as well as the initial conditions. Show the details of your work.4y" - 4y' - 3y = 0, y( -2) = e, y' (-2) = -e/ 2
Solve y" + 25y = 99 cos 4.9t, y(0) = 2, y' (0) = 0. How does the graph of the solution change if you change(a) y(0)(b) the frequency of the driving force?
Solve the IVP. Check that your answer satisfies the ODE as well as the initial conditions. Show the details of your work.y" + 4y' + (π2 + 4)y = 0, y(1/2) = 1, y' (1/2) = -2
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
Find an ODE y" + ay' + by = 0 for the given basis.e-3.1x cos 2.1x, e-3.1x sin 2.1x
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
(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.x2y'' - xy' + y = 0, y(1) = 4.3, y'
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
(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'' + 0.6y' + 0.09y = 0, y(0) =
Find the transient motion of the mass–spring system modeled by the ODE. Show the details of your work.(D2+ I)y = 5e-t cos t
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 2.9.y" + 2k2y' + k4y = 0
In a straight-line motion, let the velocity be the reciprocal of the acceleration. Find the distance y(t) for arbitrary initial position and velocity.
Find the transient motion of the mass–spring system modeled by the ODE. Show the details of your work.(D2+ 2D + 5I)y = 4 cos t + 8 sin t
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 2.9.y" + 9y' + 20y = 0
Apply the given operator to the given functions. Show all steps in detail.(D2 + 3.0D + 2.5I)y = 0
Reduce to first order and solve, showing each step in detail.y'' + (1 + 1/y)y'2 = 0
Find the transient motion of the mass–spring system modeled by the ODE. Show the details of your work.y" + 16y = 56 cos 4t
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 2.9.100y" + 240y' + (196π2 + 144)y = 0
Apply the given operator to the given functions. Show all steps in detail.(D2 + 4.80D + 5.76I)y = 0
Reduce to first order and solve, showing each step in detail.y'' = 1 + y'2
Find the transient motion of the mass–spring system modeled by the ODE. Show the details of your work.2y" + 4y' + 6.5y = 4 sin 1.5t
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 2.9.y" + y' + 3.25y = 0
Apply the given operator to the given functions. Show all steps in detail.(D2 + 3I)y = 0
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 2.9.10y" - 32y' + 25.6y = 0
Apply the given operator to the given functions. Show all steps in detail.(D2 + 4.00D + 3.36I)y = 0
Reduce to first order and solve, showing each step in detail.xy'' + 2y' + xy = 0, y1 = (cos x)/x
Reduce to first order and solve, showing each step in detail.2xy'' = 3y'
Find the steady-state motion of the mass–spring system modeled by the ODE. Show the details of your work.y" + 2.5y' + 10y = -13.6 sin 4t
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 2.9.y" + 4y' + (π2 + 4)y = 0
Apply the given operator to the given functions. Show all steps in detail.(D + 6I)2; 6x + sin 6x, xe-6x

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