Differential Equation: Laplace question

[Cornel]

Hello,

What to do in the case of blue rectangle if someone wants to solve this differential equation using laplace?

[Cornel]

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[Cornel]

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[Cornel]

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[Cornel]

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[Cornel]

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[Cornel]

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[Cornel]

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[Cornel]

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[Cornel]

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[Cornel]

@ttokoro  do you have any idea on how these problems can be solved using Laplace transform?

[ttokoro]

My Prime 9 shows same results. It needs we must define the y(x) and then it can solve.

[Cornel]

For example:

...........

Also, I do not understand then why we cannot find the solution of this kind of differential equations using this technique of Laplace Transform:

But Mathcad Prime algorithm, in this case, can find the solution:

[LucMeekes]

The Laplace transform is known (= there exists a symbolic expression) for a large number of functions and operations.

But the set of possible functions is infinitely larger. 

Example: while the Laplace transform of d/dt {y(t)} is s Y(s) + y(0), where Y(s) is the Laplace transform of y(t), there is no symbolic expression for the Laplace transform of 1/y(t).

Not every differential equation can be solved over the Laplace transform.

Success!

Luc

[Cornel]

And then on what other general method (other than Laplace Transform) I can rely in order to be able to solve these kind of differential equations that I have posted in this topic? Because we have seen that Mathcad Prime 9 algorithm somehow can solve (at least the first differential equation that I have posted), but with Laplace Transform we fail to get a solution. I thought that Laplace Transform is the general method for solving differential equations (whatever type of differential equation is involved)...

[LucMeekes]

In general a proof of correctness is found by inserting the function into the differential equation and show that it is met.

It's the general method of mathematically proving any math solutions:

10/5=2.     Why?   Because 2 * 5  = 10.

Success!

Luc

[Cornel]

Yes, but right now it is not about how to verify the solution of a differential eq (that was not the question), but instead how to find the solution of a differential eq?...If we already have a solution of a differential eq then we can insert it, but what to insert into a differential eq if we do not have any solution at our hand for that differential eq? (because we need to find at least one solution for that differential eq, in order to insert it into that differential eq), and what to do when we do not have at our hand the solution of differential eq? We try to guess at least one solution of that differential eq?

[LucMeekes · Accepted solution]

In some cases the solution was found earlier: https://en.wikipedia.org/wiki/Airy_function, https://en.wikipedia.org/wiki/Bessel_function , https://en.wikipedia.org/wiki/Laguerre_polynomials,

Sometimes its hard work, to find the solution: https://community.ptc.com/t5/Mathcad/Parabola-Mirror-an-error/m-p/365093#M142945 

Sometimes it's tougher: https://community.ptc.com/t5/Mathcad/asymptote-odesolve/m-p/183281#M72071

Maybe you find what you are looking for here: https://www-users.cse.umn.edu/~olver/ln_/odq.pdf

Success!
Luc

[Werner_E]

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@Cornel wrote:

.....

I thought that Laplace Transform is the general method for solving differential equations (whatever type of differential equation is involved)...

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Really!? So be prepared for a great disappointment. Only very few DE's are exact solvable and there is no "one method for all".  There are different methods and tricks for different types of ODE's. Laplace of course can only be used if the Laplace transforms are known. Most real world DE's are so complicated that they only can be solved numerically.

Its similar to integrals (no surprise if you consider that solving a differential equation at one step or another needs integration, likewsie the Laplace transform) - you probably know that not all integrals can be solved symbolically and that there is not THE one and only method to solve integrals. Some types of integrals can be solved using different methods like partial integration, clever types of substitutions, etc. but the majority of functions can't be integrated at all. Thats completely differently compared to calculating the derivative of a function - here we have a set of rules which, when correctly applied should be able to get the derivative of any function if the function is differentiable at all.

An then - you are using a software with a very weak symbolic engine which sure does not not all the tricks, methods and algorithms and often fails to provide a solution even though other software is able to do.

[LucMeekes]

I see you are using Prime 9. One of the new features of Prime 9 is that it is capable of solving ODE's symbolically.

I suggest you try that, instead of going the Laplace route. If Prime is capable enough, it should be able to give the solution to your ODE as

Where c is a constant.

Proof:

Success!
Luc

[Cornel]

Prime 9 result:

I want to have also the result with laplace transform (if it is possible)...as a second check

or when:

[LucMeekes]

I find the result interesting.

What number could _z be if it is not part of the set of complex numbers?

Success!

Luc

[Werner_E]

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@LucMeekes wrote:

I find the result interesting.

What number could _z be if it is not part of the set of complex numbers?

Success!

Luc

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🙂

"simplify" helps in this case