Are there any practical applications of Laplace transform? I would not use Laplace transforms to solve first, second-order ordinary differential equations as it is much easier by other methods even if it has a pulse forcing function. How can Laplace transforms be introduced so that students are motivated to learn? It needs to have an impact. What are the applications of Laplace transforms?
In a classical control theory course, Laplace transforms are used to calculate transfer functions. As suggested by @Isaiah, see what your colleagues are doing. Skipping material can be risky when other courses assume yours has covered it.
@Graham's answer goes into more detail about applications. It's good to be familiar with them so you can mention them, but maybe only in passing as motivation. If you're teaching an engineering mathematics course, it may not necessarily be your role to cover the applications themselves, or at least not all of them - your colleagues will go over them in detail in their own courses. Again, to strike the right balance, have a chat about what their topics are and what they'd like you to do in your course as mathematical preparation.
In addition, familiarity with Laplace transforms can pave the way to understanding other integral transforms, such as the z-transform and Fourier transform, which students will probably meet in courses on signal processing or partial differential equations.
"I would not use" is the incorrect framework. You need to teach your students, most efficiently, what they need as a service for their courses. Not what you like/don't.
I actually think it's a gentler way (more progressive pedagogy) to learn the topic to just do some problems, do some calculations, etc. before dealing with applications. Applications are word problems. Word problems are hard. And for LT, the applications are things like electrical circuits and mechanical controls that are not as easy to understand as a canon ball parabola or the like (not out of the world crazy, but still a little more abstract).
Take a look at the approach in Kreyszig (chapter five in the fifth edition). It's a pretty standard approach, you'd get in any ODE standalone text. Very tiny amount of motivation (not a physical problem driver like how PDE texts introduce methods). Then a bunch of techniques, the use of the tables, the partial fractions, etc. Few applications mixed in along the way and at the end. But emphasis on developing skill in the manipulations first (sans physical problems) within most sections.
You should know/figure out the applications of LT. They are classic ones and in almost all books. The textbook should inform you more than we do. FWIW, even though they like to list some mechanical applications, the honest thing for engineers is that that LT is really more important for EEs and Systems Engineers. Not so much for mechanical (and all the mechanical related ones like aero, chem, etc.) This is not to say the mechies don't have it required also. Just they won't really see it a lot later on...just one of those things they have to do as part of their general engineering.
P.s. See this related question: Should I teach Laplace Transforms? How much?
Laplace Transforms treat discontinuous forces in a natural fashion. To solve problems with discontinuous forcing functions without the transform would require tedious fitting of solutions from one region to the next at each point of discontinuity. In contrast, the Laplace transform smooths all these into one algebraic problem which once solved gives the solution as a simple inverse transform. In addition, the method allows the introduction of the transfer function which is interesting in its own right. I challenge your assertion that it is easier to work problems without the method. Even continuous problems have nice solutions with the transform. For example, it allows the derivation of the multipliers in the repeated root solution.
To me, the utility I sketch above motivates the use. But, there is a way to motivate it by an appeal to its similarity to an idea in signal processing. That is not something for which I currently have intuition. There is a well-known MIT DEqns professor who spends about 20 minutes explaining the meaning of the transform. If you search for it, you'll find it. Sorry I am lazy at the moment.
Modelling physical or electrical behaviour
Mass-spring-damper, resistor-capacitor-inductor or pump-tank-pipe systems can all be modelled using the Laplace transform. On its own this allows students to solve practical problems such as setting up a car suspension. Setting up a damper for critical damping is the obvious first step, but you can also step between Laplace-domain and time-domain work to choose a spring rate which will limit the suspension excursion to x centimetres (to prevent bottoming-out the suspension).
State-space notation allows you to also model hysteresis and time delays in systems, but state-space inherently builds on Laplace and cannot be understood before you have already grasped Laplace.
Classical control theory
Classical control theory uses PID controllers, phase lead-lag networks, and pole-zero calculations, in conjunction with models of the physical or electrical systems being controlled. All of this is 100% tied into Laplace, and nothing else is a suitable alternative. It is literally not possible to discuss classical control theory without Laplace.
IIR filters are designed for particular frequency responses. These always use Laplace, because substituting "s = jw" in the Laplace transfer function gives you the frequency response. This is relevant for all engineering disciplines, as well as anything involving audio. These filters are the building blocks for most practical filters, and understanding their behaviour is key to understanding the benefits (and drawbacks) of digital filtering with FIRs.