- Solve the IVPs by the Laplace transform. If necessary, use partial fraction expansion as in Example 4 of the text. Show all details.y' + 5.2y = 19.4 sin 2t, y(0) = 0
- Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.3t + 12
- Explain and compare the different roles of the two shifting theorems, using your own formulations and simple examples. Give no proofs.
- Make a draft of these four operations from memory. Then compare your draft with the text and write a 2- to 3-page report on these operations and their significance in applications.
- State the Laplace transforms of a few simple functions from memory.
- Consider a vibrating system of your choice modeled byy" + cy' + ky = δ(t).(a) Using graphs of the solution, describe the effect of continuously decreasing the damping to 0, keeping k constant.(b)
- Find:1 * 1
- Showing the details of your work, find L(f) if f(t) equals:3t sinh 4t
- What are the steps of solving an ODE by the Laplace transform?
- Find:1 * sin ωt
- Solve the IVPs by the Laplace transform. If necessary, use partial fraction expansion as in Example 4 of the text. Show all details.y" - y' - 6y = 0, y(0) = 11, y' (0) = 28
- Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.cos πt
- 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.t -2 (t > 2)
- Showing the details of your work, find L(f) if f(t) equals:1/2 te-3t
- Find:et * e-t
- Find and graph or sketch the solution of the IVP. Show the details.y" + 4y = δ(t - π), y(0) = 8, y’ (0) = 0
- Find and graph or sketch the solution of the IVP. Show the details.y" + 4y’ + 5y = [1 - u(t - 10)]et - e10δ(t - 10), y(0) = 0, 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.sin t(π/2 <
- Find and sketch or graph f(t) if L(f) equals(1 + e-2π(s+1))(s + 1)/((s + 1)2 + 1)
- Showing details, find f(t) if L(f) equals:
- Using Theorem 3, find f (t) if L(F) equals:
- Solve by the Laplace transform, showing the details and graphing the solution:y'1 = y2 + u(t - π), y'2 = -y1 + u(t - 2π), y1(0) = 1, y2(0) = 0
- What properties of matrix multiplication differ from those of the multiplication of numbers?
- Are there any linear systems without solutions? With one solution? With more than one solution? Give simple examples.
- Explain the use of matrices in linear transformations.
- What is the role of rank in connection with solving linear systems?
- Using the Laplace transform and showing the details of your work, solve the IVP:y''1 = -y1 + 4y2, y'2 = 3y1 - 2y2, y1(0) = 3, y2(0) = 4
- Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.cos2 ωt
- What property of the Laplace transform is crucial in solving ODEs?
- Solve the IVPs by the Laplace transform. If necessary, use partial fraction expansion as in Example 4 of the text. Show all details.y" - 1/4y = 0, y(0) = 12, y' (0) = 0
- Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.e2t sinh t
- 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.et (0 < t
- Showing the details of your work, find L(f) if f(t) equals:t cos ωt
- Find and graph or sketch the solution of the IVP. Show the details.y" + y = δ(t - π) - δ(t - 2π), y(0) = 0, y’ (0) = 1
- Using the Laplace transform and showing the details of your work, solve the IVP:y'1 = y2 + 1 - u(t - 1), y'2 = -y1 + 1 - u(t - 1), y1(0) = 0, y2(0) = 0
- When and how do you use the unit step unction and Dirac’s delta?
- Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.sin (ωt + θ)
- Solve the IVPs by the Laplace transform. If necessary, use partial fraction expansion as in Example 4 of the text. Show all details.y" + 7y' + 12y = 21e3t, y(0) = 3.5, y' (0) = -10
- 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.e-πt (2 < t
- Showing the details of your work, find L(f) if f(t) equals:t2 cosh 2t
- If you know f(t) = L-1{F(s)}, how would you find L-1{F(s)/s2}?
- Find and graph or sketch the solution of the IVP. Show the details.4y" + 24y’ + 37y = 17e-t + δ(t - 1/2), y(0) = 1, y’ (0) = 1
- Using the Laplace transform and showing the details of your work, solve the IVP:y'1 = 2y1 - 4y2 + u(t - 1)et, y2 = y1 - 3y2 + u(t - 1)et, y1(0) = 3, y2(0) = 0
- Showing the details of your work, find L(f) if f(t) equals:te-kt sin t
- Explain the use of the two shifting theorems from memory.
- 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' + 3y = 6t - 8, y(0) = 0, y' (0) = 0
- Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.
- 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 (t > 3/2)
- Showing the details of your work, find L(f) if f(t) equals:1/2t2 sin πt
- Can a discontinuous function have a Laplace transform? Give reason.
- Solve by the Laplace transform, showing the details:
- Using the Laplace transform and showing the details of your work, solve the IVP:y'1 = 4y1 + y2, y2 = -y1 + 2y2, y1(0) = 3, y2(0) = 1
- If two different continuous functions have transforms, the latter are different. Why is this practically important?
- Using the Laplace transform and showing the details of your work, solve the IVP:y"1 = y1 + 3y2, y2" = 4y1 - 4et, y1(0) = 2, y'1(0) = 3, y2(0) = 1, y'2(0) = 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" + 3y' + 2.25y = 9t3 + 64, y(0) = 1, y' (0) = 31.5
- Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants. 19
- Showing the details of your work, find L(f) if f(t) equals:4t cos 1/2πt
- Find the transform, indicating the method used and showing the details.5 cosh 2t - 3 sinh t
- Find and graph or sketch the solution of the IVP. Show the details.y" + 5y’ + 6y = u(t - 1) + δ(t - 2), y(0) = 0, y’ (0) = 1
- Solve by the Laplace transform, showing the details:
- Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.
- Using the Laplace transform and showing the details of your work, solve the IVP:y"1 + y2 = -101 sin 10t, y2" + y1 = 101 sin 10t, y1(0) = 0, y'1 (0) = 6, y2(0) = 8, y'2 (0) = -6
- Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.
- Solve the shifted data IVPs by the Laplace transform. Show the details.y' - 6y = 0, y(-1) = 4
- Find and sketch or graph f(t) if L(f) equals6(1 - e-πs)/(s2 + 9)
- Find the transform, indicating the method used and showing the details.sin2 (1/2 πt)
- Experiment with the graphs of l0, . . , l10, finding out empirically how the first maximum, first minimum, . . . is moving with respect to its location as a function of n. Write a short report on
- (a) Show that for a simple root α and fraction A/(s - α) in F(s)/G(s) we have the Heaviside formula (b) Similarly, show that for a root α of order m and fractions in we have the Heaviside
- Solve by the Laplace transform, showing the details:
- Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.
- Using the Laplace transform and showing the details of your work, solve the IVP:y'1 + y'2 = 2 sinh t, y'2 + y'3 = et, y'3 + y'1 = 2et + e-t, y1(0) = 1, y2(0) = 1, y3(0) = 0
- Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.
- Solve the shifted data IVPs by the Laplace transform. Show the details.y" + 3y' - 4y = 6e2t-3, y(1.5) = 4, y' (1.5) = 5
- Find and sketch or graph f(t) if L(f) equalse-3s/s4
- Find the transform, indicating the method used and showing the details.et/2u(t - 3)
- Using differentiation, integration, s-shifting, or convolution, and showing the details, find f(t) if L(t) equals:
- Find the Laplace transform of the staircase function in Fig. 140 by noting that it is the difference of kt/p and the function in 14(d).
- (a) Replace 2 in Prob. 13 by a parameter k and investigate graphically how the solution curve changes if you vary k, in particular near k = -2. Data from Prob. 13 (b) Make similar experiments with
- Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants. 1 2
- In Prob. 16, vary the initial conditions systematically, describe and explain the graphs physically. The great variety of curves will surprise you. Are they always periodic? Can you find empirical
- Convert table 6.1 to a table for finding inverse transforms (with obvious changes, e.g., L-1(1/sn) = tn-1/(n - 1), etc).
- Find the transform, indicating the method used and showing the details.t cos t + sin t
- Using differentiation, integration, s-shifting, or convolution, and showing the details, find f(t) if L(t) equals:
- Using Laplace transforms, find the currents i1(t) and i2(t) in Fig. 148, where v(t) = 390 cos t and i1(0) = 0, i2(0) = 0. How soon will the currents practically reach their steady state?
- Using (1) or (2), find L(f) if f(t) if equals:sin2 ωt
- Derive formula 6 from formulas 9 and 10.
- Find the transform, indicating the method used and showing the details.12t * e-3t
- Using the Laplace transform and showing the details, solvey" + 6y’ + 8y = e-3t - e-5t, y(0) = 0, y’ (0) = 0
- Using differentiation, integration, s-shifting, or convolution, and showing the details, find f(t) if L(t) equals:
- Showing details, find f(t) if L(f) equals:
- Give simple examples of functions (defined for all t > 0) that have no Laplace transform.
- Find the inverse transform, indicating the method used and showing the details:
- Using the Laplace transform and showing the details, solvey" + 9y = 8 sin t if 0 < t < π and 0 if t > π; y(0) = 0, y’ (0) = 4
- Find the inverse transform, indicating the method used and showing the details:
- Using the Laplace transform and showing the details, solvey" + y’ - 2y = 3 sin t - cos t if 0 < t < 2π and 3 sin 2t - cos 2t if t > 2π; y(0) = 1, y’ (0) = 0
- If L(f(t)) = F(s) and c is any positive constant, show that L(f(ct)) = F(s/c)/c (Use (1).) Use this to obtain L(cos ωt) from L(cos t).
- Find the inverse transform, indicating the method used and showing the details:
- Using the Laplace transform and showing the details, solvey" + y = t if 0 < t < 1 and 0 if t > 1; y(0) = 0, y’ (0) = 0
- Given F(s) = L(f), find f(t). a, b, L, n are constants. Show the details of your work.
- Find the inverse transform, indicating the method used and showing the details: