What types of ODEs can Laplace Transforms effectively solve?

[ice109]

what class of ODE problems does a laplace transform solve? it seems like it solves all first and second order problem with constant coefficients and variables coefficients require series solutions.

 

[Marin]

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:

it seems like it solves all first and second order problem with constant coefficients and variables coefficients require series solutions.

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

 

[ice109]

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

aww that sucks. admittedly i was hoping to skip relearning all those convoluted methods and just learn the laplace really well.

 

[Marin]

A couple of more methods is after all not so bad I think :) - Laplace transform is very useful when you have the initial conditions, because it produces for you straightaway the general solution to an ODE, with all the constants and s.o. :)

It's a powerful technique ;)

 

[Defennder]

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).

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).