it seems like it solves all first and second order problem with constant coefficients and variables coefficients require series solutions.
Marin said:Hello, ice109!
I have learned Laplace Transform by myself so I am feeling a bit like an amateur talking about it here in the forum, but I think we must add something to the statement above:
If you're talking about homogeneous ODEs this is, I suppose, correct. But if the ODE is inhomogeneous, then it is exactly this inhomogeneous part which determines, if the equation can be Laplace-transformed:
e.g.: y''[x]+y'[x]+y[x] = tanx
This 2nd order ODE cannot be solved via Laplace transform, since the Laplace transform for tanx is not 'broadly' defined (if defined at all). I mean I haven't seen it in the tables and the integral needed to solve using the definition of the transform does not produce an elementary function (which we need).
best regards, Marin
Yeah, it doesn't exist for tan x, sec x, csc x, cot x or any other function which is not piecewise continuous as well those functions which diverge at a greater rate than e^(at).Marin said:e.g.: y''[x]+y'[x]+y[x] = tanx
This 2nd order ODE cannot be solved via Laplace transform, since the Laplace transform for tanx is not 'broadly' defined (if defined at all). I mean I haven't seen it in the tables and the integral needed to solve using the definition of the transform does not produce an elementary function (which we need).
A Laplace transform is a mathematical operation that transforms a function of time into a function of complex frequency. It is commonly used in engineering and physics to solve differential equations.
A Laplace transform can convert the problem of solving a differential equation into a simpler algebraic equation, making it easier to solve using standard mathematical techniques.
Using a Laplace transform allows for the solution of differential equations that may be difficult or impossible to solve using traditional methods. It also provides a systematic approach for solving ODEs and can handle a wide range of initial conditions.
One limitation of using Laplace transform is that it may not work for all types of differential equations. It is most effective for linear, time-invariant systems and may not yield a solution for nonlinear or time-varying systems.
The main difference between using a Laplace transform and other methods is that it transforms the problem into the frequency domain, making it easier to solve using algebraic equations. Other methods, such as Euler's method or Runge-Kutta method, solve the equations directly in the time domain.