What is the most useful method to solve differential equations?

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Summary:

I really love Laplace transformations. I was able to solve first and second order differential equations easily while having fun. I’m not so sure why but it just came to me naturally.

Main Question or Discussion Point

What is yours?

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Summary:: I really love Laplace transformations. I was able to solve first and second order differential equations easily while having fun. I’m not so sure why but it just came to me naturally.

What is yours?

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jedishrfu
Mentor
The best is to hand it to someone else to solve. :-)

WWGD
Gold Member
2019 Award
The best is to hand it to someone else to solve. :-)
I call it teamwork: Others do the work, I take the credit ;).

HallsofIvy
Homework Helper
The best is to hand it to someone else to solve. :-)
Isn't that what this forum is for?
Anonymous, there is no one method that will apply to all differential equations. Even the Laplace transform method only applies to linear differential equations with constant coefficients. And I think there are other methods (characteristic equation for the associated homogeneous equation, "variation of parameters" or "undetermined coefficients") for those that work better than Laplace transform.

Popular method is "ask AI friend for help". Mathematica is knowledgeable and willing to help

MATHEMATICA

jasonRF
Gold Member
Popular method is "ask AI friend for help". Mathematica is knowledgeable and willing to help
A related technique is looking it up in a book. This was a go-to method in graduate school many years ago - how else would I know the solution was a parabolic cylinder function, or some other uncommon beast?

Staff Emeritus
2019 Award
What is the most useful method to solve differential equations?
Assume a solution of the form...

epenguin
Homework Helper
Gold Member
Assume a solution of the form...
In other words: know the answer. At least in rough outline, then you can hammer the rough form into an exact fit.

I have often thought that presenting differential equations as a 'subject' that has 'methods' of 'solving' is a bit of a fraud.
The same thing for that matter for simple integration of functions.

That sometimes books are more honest, perhaps this is intentional by the authors, when excercises say "find solutions" rather than "solve".

I think if I were teaching to people not mathematicians but heavy maths users like engineers, chemists etc. I would follow any equation of the subject of interest with the solution, which the students can verify by differentiation, which is a genuine subject they should have learned (and which by the way we are continually telling them to do when they come here asking whether their solutions are correct) instead of pretending to 'solve' it. And advise them to do the same in their reading or studying their subject.

jasonRF
Gold Member
Assume a solution of the form...
I had a professor that called this "the method of guessing." So that is what I usually call it.

jasonRF
Gold Member
Back to the OP, the method of Frobenius is pretty nice. I recall it being tedious when I learned it, but (25 years after taking the class!) actually had a need to use it for my work fairly recently and thought it was kind of fun. It allowed me to compute the solution of a "simple" problem to any accuracy I wanted, so that I could use it as a test case to validate a numerical simulation I will be using to solve problems that are impossible to solve analytically.

Ssnow
Gold Member
For partial differential equation there is '' the method of characteristics ''. It is very useful (typically it applies to first order equations).

Ssnow