- 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
- 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
- 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
- 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,
- 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
- 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
- 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,
- 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
- 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) =
- 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 =
- 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,
- 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 =
- Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.
- 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,
- 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
- Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.
- 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) =
- 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
- (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
- 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) =
- 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
- Find the transform. Show the details of your work. Assume that α, b, ω, θ are constants.
- In Prob. 16, vary the initial conditions systematically, describe and explain the graphs physically. The great variety of curves will surprise you.
- 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
- 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;
- 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: