The discussion centers on the types of ordinary differential equations (ODEs) that can be effectively solved using Laplace transforms. Participants explore the applicability of Laplace transforms to first and second order ODEs, particularly focusing on the conditions under which these transforms can be utilized, including considerations of homogeneous versus inhomogeneous equations.
Participants generally agree that Laplace transforms are effective for certain classes of ODEs, particularly those with constant coefficients. However, there is disagreement regarding the limitations imposed by inhomogeneous terms and specific functions, indicating that the discussion remains unresolved on the broader applicability of Laplace transforms.
Limitations noted include the dependence on the nature of the inhomogeneous part of the ODE and the specific functions involved, which may not be suitable for Laplace transformation.
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).