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advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

In Probs. 37€“45 find the inverse transform. Show the details of your work. (s + 1)3
Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 129, assuming zero initial current and charge on the capacitor and:L = 1 H, C = 0.25 F, v = 200 (t -
Solve by the Laplace transform, showing the details and graphing the solution:y'1 = 2y1 + 4y2, y'2 = y1 + 2y2, y1(0) = -4, y2(0) = -4
Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 128 with R = 10 Î© and C = 10-2F, where the current at t = 0 is assumed to be zero,
Solve by the Laplace transform, showing the details and graphing the solution:y'1 = y2, y'2 = -4y1 + δ(t - π), y1(0) = 0, y2(0) = 0
Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 128 with R = 10 Î© and C = 10-2F, where the current at t = 0 is assumed to be zero, and:v
Solve by the Laplace transform, showing the details and graphing the solution:y" + 4y = δ(t - π) - δ(t - 2π), y(0) = 1, y' (0) = 0
Given F(s) = L(f), find f(t). a, b, L, n are constants. Show the details of your work. (s + a)(s + b)
Solve by the Laplace transform, showing the details and graphing the solution:y" + 16y = 4δ(t - π), y(0) = -1, y' (0) = 0
Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 126, assuming i(0) = 0 and:R = 10 Î©, L = 0.5 H, v = 200t V if 0 < t < 2 and 0 if t
(a) Give reasons why Theorems 1 and 2 are more important than Theorem 3.(b) Extend Theorem 1 by showing that if f(t) is continuous, except for an ordinary discontinuity (finite jump) at some t =
Given F(s) = L(f), find f(t). a, b, L, n are constants. Show the details of your work. 1 (s + V2(s – V3)
Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 126, assuming i(0) = 0 and:R = 1 kÎ© (= 1000 Î©), L = 1 H, v = 0 if 0 < t
Find the inverse transform, indicating the method used and showing the details: 3s s2 - 2s + 2
Using the Laplace transform and showing the details, solvey" + 2y’ + 5y = 10 sin t if 0 < t < 2π and 0 if t > 2π; y(π) = 1, y’ (π) = 2e-π - 2
Find the inverse transform, indicating the method used and showing the details: 2s – 10 -5s
Showing details, find f(t) if L(f) equals: 240 (s2 + 1)(s + 25)
Prove that L-1 is linear. Use the fact that L is linear.
Using the Laplace transform and showing the details, solvey" + 3y’ + 2y = 1 if 0 < t < 1 and 0 if t > 1; y(0) = 0, y’ (0) = 0
Find the inverse transform, indicating the method used and showing the details: 2 - 6.25 (s2 + 6.25)2
Using Theorem 3, find f (t) if L(F) equals: 20 g3 – 2Ts?
Using the Laplace transform and showing the details, solvey" + 3y’ + 2y = 4t if 0 < t < 1 and 8 if t > 1; y(0) = 0, y' (0) = 0
Show that L(1/√t) = √π/s. [Use (30) Г(1/2) = √π in App. 3.1.] Conclude from this that the conditions in Theorem 3 are sufficient but not necessary for the existence of a Laplace transform.
Find the inverse transform, indicating the method used and showing the details: 16 s2 + s+
Proceeding as in Example 1, obtain(a) L(t cos ωt) = S2 - ω2/(s2 + ω2)2and from this and Example 1(b) formula 21(c) 22(d) 23 in Sec. 6.9(e) L(t cosh αt) = S2 + α2/(s2 - α2)2(f) L(t sinh
Using differentiation, integration, s-shifting, or convolution, and showing the details, find f(t) if L(t) equals: s+a In
Using the Laplace transform and showing the details, solvey" + 10y’ + 24y = 144t2, y(0) = 19/12, y’ (0) = -5
Find the inverse transform, indicating the method used and showing the details: 7.5 s2 - 2s - 8
Show that et2 does not satisfy a condition of the form (2).
Using (1) or (2), find L(f) if f(t) if equals:sin4 t. Use Prob. 19.Data from Prob. 19sin2 ωt
Solve Prob. 19 when the EMF (electromotive force) is acting from 0 to only. Can you do this just by looking at Prob. 19, practically without calculation?Data from Prob. 19Using Laplace transforms,
Using differentiation, integration, s-shifting, or convolution, and showing the details, find f(t) if L(t) equals:arccot s/π
Using the Laplace transform and showing the details, solve9y" - 6y' + y = 0, y(0) = 3, y’ (0) = 1
Find the transform, indicating the method used and showing the details.(sin ωt) * (cos ωt)
Using L(f) in Prob. 10, find L(f1), where f1(t) = 0 if t<2 and f1(t) = 1 if t > 2.Data from Prob. 10Find the transform. Show the details of your work. Assume that Î±, b,
Using (1) or (2), find L(f) if f(t) if equals:cos2 2t
Prove:(a) Commutativity, f * g = g * f(b) Associativity, ( f * g) * v = f * (g * v)(c) Distributivity, f * (g1 + g2) = f * g1 + f * g2(d) Dirac€™s delta. Derive the sifting formula (4) in
Using differentiation, integration, s-shifting, or convolution, and showing the details, find f(t) if L(t) equals: 2s + 6 (s2 + 6s + 10)2
Find the transform, indicating the method used and showing the details.u(t - 2π) sin t
Find and sketch or graph f(t) if L(f) equals2(e-s - e-3s)/(s2 - 4)
Using (1) or (2), find L(f) if f(t) if equals:t cos 4t
Solve by the Laplace transform, showing the details: У) + 2e| ут)е"" dт 3 te'
(a) The Laplace transform of Î± piece wise continuous function f(t) with period p isProve this theorem. Write ˆ«ˆž0 = ˆ«p0 + ˆ«2pp + . .
Using differentiation, integration, s-shifting, or convolution, and showing the details, find f(t) if L(t) equals: (s2 + 16)2
Find the transform, indicating the method used and showing the details.16t2u(t - 1/4)
Find and sketch or graph f(t) if L(f) equals4(e-2s - 2e-5s)/s
Solve the shifted data IVPs by the Laplace transform. Show the details.y" + 2y' + 5y = 50t - 100, y(2) = -4, y' (2) = 14
Using the Laplace transform and showing the details of your work, solve the IVP:4y'1 + y2r - 2y'3 = 0, -2y'1 + y'3 = 1, 2y'2 - 4y'3 = -16t
Solve by the Laplace transform, showing the details: y(7) cosh (t – T) dr = t + e yt) +
Find and graph or sketch the solution of the IVP. Show the details.y" + 2y’ + 5y = 25t - 100δ(t - π), y(0) = -2, y’ (0) = 5
Find the transform, indicating the method used and showing the details.e-t(cos 4t - 2 sin 4t)
Show that
Find and sketch or graph f(t) if L(f) equalse-3s/(s - 1)3
Solve the shifted data IVPs by the Laplace transform. Show the details.y" - 2y' - 3y = 0, y(4) = -3, y' (4) = -17
Using the Laplace transform and showing the details of your work, solve the IVP:y"1 = -2y1 + 2y2, y2" = 2y1 - 5y2, y1(0) = 1, y'1 (0) = 0, y2(0) = 3, y'2 (0) = 0
Using the Laplace transform and showing the details of your work, solve the IVP:y'1 = -y2, y'2 = -y1 + 2[1 - u(t - 2π)] cos t, y1(0) = 1, y2(0) = 0
Solve by the Laplace transform, showing the details: yt) – y(7) sin 2(t – T) dr = sin 2t
Find and graph or sketch the solution of the IVP. Show the details.y" + 5y’ + 6y = δ(t - 1/2π) + u(t - π) cos t, y(0) = 0, y’ (0) = 0
Showing the details of your work, find L(f) if f(t) equals:tnekt
Sketch or graph the given function, which is assumed to be zero outside the given interval. Represent it, using unit step functions. Find its transform. Show the details of your work.sinh t (0 < t
Find the transform. Show the details of your work. Assume that Î±, b, Ï‰, Î¸ are constants. k
Solve the IVPs by the Laplace transform. If necessary, use partial fraction expansion as in Example 4 of the text. Show all details.y" + 0.04y = 0.02t2, y(0) = -25, y' (0) = 0
Using the Laplace transform and showing the details of your work, solve the IVP:y'1 = -2y1 + 3y2, y'2 = 4y1 - y2, y1(0) = 4, y2(0) = 3
Solve by the Laplace transform, showing the details: y(t) + 4 y(7)(t – 7) dr = 2t 0.
Find and graph or sketch the solution of the IVP. Show the details.y" + 3y’ + 2y = 10(sin t + δ(t - 1)), y(0) = 1, y’ (0) = -1
Sketch or graph the given function, which is assumed to be zero outside the given interval. Represent it, using unit step functions. Find its transform. Show the details of your work.t2 (1 < t
Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.1.5 sin (3t - π/2)
Solve the IVPs by the Laplace transform. If necessary, use partial fraction expansion as in Example 4 of the text. Show all details.y" - 4y' + 4y = 0, y(0) = 8.1, y' (0) = 3.9
Using the Laplace transform and showing the details of your work, solve the IVP:y'1 = 5y1 + y2, y'2 = y1 + 5y2, y1(0) = 1, y2(0) = -3
Find and graph or sketch the solution of the IVP. Show the details.y" + 4y’ + 5y = δ(t - 1), y(0) = 0, y’ (0) = 3
Showing the details of your work, find L(f) if f(t) equals:t2 sin 3t
Sketch or graph the given function, which is assumed to be zero outside the given interval. Represent it, using unit step functions. Find its transform. Show the details of your work.sin πt (2 <
Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.e-t sinh 4t
Solve the IVPs by the Laplace transform. If necessary, use partial fraction expansion as in Example 4 of the text. Show all details.y" - 6y' + 5y = 29 cos 2t, y(0) = 3.2, y' (0) = 6.2
Use the powerful formulas (21) to do Probs. 19–25. Show the details of your work.Evaluate ∫x-1 J4(x) dx.
Use the powerful formulas (21) to do Probs. 19–25. Show the details of your work.Derive (1) from (21).
Derive (22) in Example 3 from (27).
Solve the initial value problem by a power series. Graph the partial sums of the powers up to and including x5. Find the value of the sum s (5 digits) at x1.(1 - x2)y" - 2xy' + 30y = 0, y(0) = 0, y'
Find a basis of solutions. Try to identify the series as expansions of known functions. Show the details of your work.xy" + 3y' + 4x3y = 0
Find a general solution in terms of hyper geometric functions.4(t2 - 3t + 2)ÿ - 2ý + y = 0
Find a basis of solutions. Try to identify the series as expansions of known functions. Show the details of your work.x2y" + 2x3y' + (x2 - 2)y = 0
Gauss€™s hypergeometric ODE is
ZEROS of Bessel functions play a key role in modelingCompute the first four positive zeros of J0(x) and J1(x) from (14). Determine the error and comment.
Differentiating (13) with respect to u, using (13) in the resulting formula, and comparing coefficients of un, obtain the Bonnet recursion.where n = 1, 2, . . . . This formula is useful for
Find a basis of solutions. Try to identify the series as expansions of known functions. Show the details of your work.16(x + 1)2y" + 3y = 0
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.x2y" + 6xy' + (4x2 + 6)y = 0
(a) Experiment with (14) for integer n. Using graphs, find out from which x = xn on the curves of (11) and (14) practically coincide. How does xn change with n?(b) What happens in (b) if n = ±1/2?
Modified Bessel functions of the first kind of order v are defined by Iv (x) = i-vJv (ix), i = ˆš-1. Show that Ivsatisfies the ODE
Applying the binomial theorem to (x2- 1)n, differentiating it n times term by term, and comparing the result with (11), show that
Find a power series solution in powers of x. Show the details.(1 - x2)y" - 2xy' + 2y = 0
Find a basis of solutions. Try to identify the series as expansions of known functions. Show the details of your work.xy" + (1 - 2x) y' + (x - 1) y = 0
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' + 4xy = 0
It can be shown that for large x,with ˆ¼ defined as in (14) of Sec. 5.4.(a) Graph Yn for n = 0, . . ., 5 on common axes. Are there relations between zeros of one function and extrema of
Find a power series solution in powers of x. Show the details.y" - y' + xy = 0
Generating functions play a significant role in modern applied mathematics. The idea is simple. If we want to study a certain sequence (fn(x)) and can find a functionis a generating function of the
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" + k2x4y = 0 (y = u√x, 1/3kx3 =
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" + 2x3y' + (x2 - 2)y = 0
Graph P2(x), . . . , P10(x) on common axes. For what x (approximately) and n = 2, . . . , 10 is |Pn(x)| < 1/2?
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" + y = 0
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" + xy = 0 (y = u√x, 2/3x3/2 = z)

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