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# The Laplace transform as an alternative general method for solving linear ordinary differential equations

• The Laplace transform is a popular approach in solving ordinary differential equations (ODEs), particularly solving initial value problems (IVPs) of ODEs. Such stereotype may confuse students when they face a task of solving ODEs without explicit initial condition(s). In this paper, four case studies of solving ODEs by the Laplace transform are used to demonstrate that, firstly, how much influence of the stereotype of the Laplace transform was on student's perception of utilizing this method to solve ODEs under different initial conditions; secondly, how the generalization of the Laplace transform for solving linear ODEs with generic initial conditions can not only break down the stereotype but also broaden the applicability of the Laplace transform for solving constant-coefficient linear ODEs. These case studies also show that the Laplace transform is even more robust for obtaining the specific solutions directly from the general solution once the initial values are assigned later. This implies that the generic initial conditions in the general solution obtained by the Laplace transform could be used as a point of control for some dynamic systems.

 Citation: • • Figure 1.  A system represented by a single block diagram in the state space

Figure 2.  The series RL circuit with input $f(t) = E = {E_0}\sin \omega t$

Figure 3.  Electric currents in the RL circuit with R = 10 Ω, L = 5 H, ω = 2, and E0 = 10 V

Figure 4.  Plot of the output for Case 2 with y(0) = 0 and y′(0) = 0

Figure 5.  An example of student's work on solving the ODE in Case 2 by the Laplace transform

Figure 6.  Plots of the solution to Case 3 with fixed y(0) = 1 and different values for $y'(0) = {v_1}$

Figure 7.  Electric currents of the RL circuit in Case 1 with R = 10 Ω, L = 5 H, ω = 2, and E0 = 10 V [v0 = i(0) for –10, 0, and 10 amperes, respectively]

Figure 8.  Plots of the output of the ODE in Case 2 by Laplace transform with different initial values [Black: v0 = –3, v1 = –3; Red: v0 = 0, v1 = 0; Blue: v0 = 5, v1 = 5]

Figure 9.  The mixing problem for Case 4

Figure 10.  Plots of the mixing processes in the solutions (34) and (35) V = 4000 liters, x0 = 600 kg, y0 = 0 kg; blue curves: q1 = 40 l/m; red curves: q2 = 100 l/m

Figure 11.  Plots of the mixing processes in the solutions (36) and (37) V = 4000 liters, x0 = 500 kg, y0 = 100 kg; blue curves: q1 = 40 l/m; red curves: q2 = 100 l/m

Table 1.  Summary of the performances in solving the ODE in Case 2 by students

 Laplace transform Laplace transform Convolution Integration Incorrect 2 6 3 Correct 24 24 24

Table 2.  Summary of attempts to solve Case 3 by 124 students

 $y(0) = 1$ & $y'(0) = 0$ $y(0) = 1$ & $y'(0) = {v_1}$ No attempt 105 105 Convolution 8 (6) 0 Partial fraction 11 (5) 0 Correct solution 11 0 Italic numbers indicate the correct solutions obtained by students
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