Solution of differential equations with boundary conditions contain "Limit"

Questions
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1

Dear Friends,

I am going to solve the following differential equation:

y''\[x\] = {k^2}y\[x\]

with boundary conditions is y'[0] - hy[0] = 0 and \mathop {Lim}\limits_{x \to \infty } [\left| {y\[x\]} \right| < \infty
I tried to solve it in Mathematica by

DSolve[{y''[x] == {k^2}y[x],y'[0] - hy[0] == 0,Limit[Abs[y[x]], x -> \[Infinity] < \[Infinity]]},y[x],x]

But I get the following error:

Equation or list of equations expected instead of Limit[Abs[y[x]],x->False] in the first argument.

I can use the boundary conditions contain “Limit” how to correct?

I would highly appreciate it if somebody can help me.

Best regards.

2

Syntax errors. Try following:

In[1]:= DSolve[{y’‘[x] == k^2 y[x], y’[0] - h y[0] == 0}, y[x], x]

Out[1]= {{y[x] → (
E^(-k x) (-h + E^(2 k x) h + k + E^(2 k x) k) C[1])/(h + k)}}

Clearly then Lim x->Infinity, gives y[x]->Infinity

3

Unless C[1]=0, in which case the solution is constant zero, which appears to be a valid solution.

M.

4

Thank you very much for helping me! But I still exist two problems:

  1. The boundary conditions of the problem is: \mathop {Lim}\limits_{x \to \infty } [\left| {y\[x\]} \right| < \infty. (y [x] is always finite).
  2. I can not use Plot command for y\[x\] because the answer is still one constant C.

Could I eliminate constant C by the second condition? how can i do that?

Please help me one more!! I sincerely thank you very much!

5

For a linear DE with linear BCs, there is always an arbitrary multiplicative constant in the solution. The constant is often determined by a normalization condition on f(x).

6

Well, if C[1] is not equal zero, then the solution will diverge. So if you want the limit to be finite, i.e. <\infty, then the only thing that works is C[1]=0. So using that condition of yours, the only choice for the constant that works seems to be C[1]=0.

Cheers,

M.

7

This is what I expected!! I really appreciate your help with my problem! Thank you so much!!