0
$\begingroup$

I'm trying to solve the following differential equation with Wolfram Alpha (I have a pro subscription):

y''+2*a*y'+(a^2-c^2-b/d*Exp(-x/d)-b^2*Exp(-2*x/d))*y=0

I want to assume that 'a', 'b', 'c', 'd' and 'x' are real positive numbers. How do I do this in Wolfram|Alpha?

Currently, by not assuming that, I'm getting analytic solutions but it contains associated Laguerre polynomials 'L': $$ y(x)=k_1 U\left(c d+1,2 c d+1,2 b d e^{-x/d}\right) \exp \left(d \left((a+c) \log \left(e^{-x/d}\right)-b e^{-x/d}\right)\right)+k_2 L_{-c d-1}^{2 c d}\left(2 b d e^{-x/d}\right) \exp \left(d \left((a+c) \log \left(e^{-x/d}\right)-b e^{-x/d}\right)\right) $$ where the n=-cd-1 is a not a real number.

Edit: Didn't know but Laguerre polynomials are defined for non-integer orders as well. My question still remains for the syntax of "assuming" in Wolfram Alpha though.

$\endgroup$
0
$\begingroup$

Wolfram Alpha can read Mathematica syntaxis, and so I would use the full correct expression with Assumptions, for example:

DSolve[y''[x] + 
   2*a*y'[x] + (a^2 - c^2 - b/d*Exp[-x/d] - b^2*Exp[-2*x/d])*y[x] == 
  0, y[x], x, Assumptions -> {Element[a | b | c | d, Reals]}]

However, this seems too complicated for WA.


But your equation has too many unnecessary parameters.

Let's move to a new variable:

$$t=x \cdot d, \quad y(x)=f(t)$$

And new parameters:

$$ad=A, \quad bd=B, \quad cd=C$$

Now we have:

$$f''(t)+2 A f'(t)+ \left(A^2-\text{C}^2-B^2 \exp (-2 t)-B \exp (-t)\right)f(t)=0$$

This is not all, we can also move to another variable:

$$s=t-\log B, \quad f(t)=g(s)$$

$$g''(s)+2 A g'(s)+ \left(A^2-\text{C}^2-\exp (-2 t)-\exp (-t)\right)f(t)=0$$

Note that Mathematica has capital C reserved, so I will use Cc:

DSolve[g''[s] + 
   2 A g'[s] + (A^2 - Cc^2 - Exp[-2 s] - Exp[-s]) g[s] == 0, g[s], s, 
 Assumptions -> {Element[A | B, Reals], A > 0, B > 0}]

{{g[s] -> 
   E^(-E^-s + (A + Cc) Log[E^-s])
      C[1] HypergeometricU[1 + Cc, 1 + 2 Cc, 2 E^-s] + 
    E^(-E^-s + (A + Cc) Log[E^-s])
      C[2] LaguerreL[-1 - Cc, 2 Cc, 2 E^-s]}}

Or:

$$g(s)=c_1 U\left(\text{C}+1,2 \text{C}+1,2 e^{-s}\right) e^{(A+\text{C}) \log \left(e^{-s}\right)-e^{-s}}+c_2 L_{-\text{C}-1}^{2 \text{C}}\left(2 e^{-s}\right) e^{(A+\text{C}) \log \left(e^{-s}\right)-e^{-s}}$$

Wolfram Alpha still doesn't take this, however it doesn't matter as we still get the exact same solution you've got, in terms of generalized Laguerre polynomial and confluent hypergeometric function.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.