Laplace Transform or Characteristic Equation? The proponents of the use of Laplace Transform in differential equations claim that is easier and faster but is that always the case? Often I have found out that solving an ODE through the characteristic equation and the use of undetermined coefficients is significantly easier and less time-consuming. What is your opinion? Thank you.
 A: Having just taught this course recently, my opinion.


*

*For homogeneous equations, of course, there is no difference; either way you have to factor the characteristic polynomial and without using Laplace transforms you can still just write down the exponential solutions.

*For inhomogeneous equations where the inhomogeneous term is of the form amenable to undetermined coefficients, of course, both this and Laplace transforms are still options.  Now, for an inhomogeneous term such as $t^2 e^{-3t} \sin(2t)$, both methods are going to involve significant annoyance: either lots of product rule for undetermined coefficients, or lots of quotient rule for the Laplace transform.  But, and this is pretty much a vague heuristic: in solving an equation such as
$$x^{(4)} + x = t^2 e^{-3t} \sin(2t)$$
by undetermined coefficients you'll have to take a fourth derivative using the product rule all the way, while with Laplace transforms you have only to take the second derivative of $(s + 3)/((s + 3)^2 + 4)$, which, while annoying, is not nearly as bad, and crucially, does not get any worse if the degree of the equation goes up.

The computations with undetermined coefficients get slightly better if you use complex exponentials, but I think this point remiains valid.

*This point isn't as compelling for low-order equations.  When solving equations of order 2, the Laplace transform may in fact involve more derivatives than undetermined coefficients, depending on how high the polynomial goes in the inhomogeneous term.  Of course, undetermined coefficients involves solving a pretty big system of equations in that case, but it's not actually a very hard system, while taking those derivatives really is pretty bad.

*Laplace transforms have a computational advantage in applying the initial conditions, since you don't have to pass through the general solution to figure out the unknown constants.  This turns a two-step solution process into one step: take the Laplace transform, which necessitates applying the initial conditions immediately, and then rearrange and take the inverse transform.  With undetermined coefficients you have first an awful computation to find the eponymous coefficients, and then another awful computation (of exactly the same nature) to find the unknown coefficients of the homogeneous part of the solution that fit the initial conditions.

*For inhomogeneous terms involving any other kind of function, you can't use undetermined coefficients.  That's not to say that the Laplace transform is the only option: if the function is analytic then you can use series expansion, and this is probably a lot more efficient for just getting an answer, and a well-approximated one at that.  Laplace transforms don't have any theoretical advantage in this domain of problem.

*The one place you really must use Laplace transforms is if the equation involves a distribution of some kind: a delta function, step function, or so on.  In fact, you can really only define such a differential equation using some kind of integral transform, since what else does the delta function mean?  That's what distributions are.

*If the equation doesn't have constant coefficients, you're probably better off with series methods.  But since you ask about the characteristic equation, I think this is not what concerns you.
In summary, Laplace transforms have the advantage that they provide a particular solution more easily, and they provide a separation of work between the homogeneous and inhomogeneous parts of the equation, rather than just a separation of the solution.  As for which is better: there's no answer.  Take note of the points above and see which one is the dominant factor in your problem, and of course, get some experience (I can't say I have that much experience myself, but with some general mathematical experience, I can make more of what I've seen).
A: Note that, Laplace transform technique is efficient for solving systems of linear ordinary differential equations with constant coefficients. 
