Re: A ODE I need to solve

• To: mathgroup at smc.vnet.net
• Subject: [mg111105] Re: A ODE I need to solve
• From: schochet123 <schochet123 at gmail.com>
• Date: Tue, 20 Jul 2010 03:44:58 -0400 (EDT)
• References: <i20q77\$kh9\$1@smc.vnet.net>

```On Jul 19, 9:09 am, Sam Takoy <sam.ta... at yahoo.com> wrote:
> Hi,
>
> Mathematica has a problem with this:
>
> DSolve[y''[x] + (2/Cosh[x - h]^2 - 1) y[x] == 0, y, x]
>
> although the solution is not too difficult. One of the solutions is
>
> 1/(Cosh[2(x-h)]+1)^(1/2)
>
> Is there a way to help Mathematica along?
>
> Thanks!

Define

lhs[y_] := D[y, x, x] + (2/Cosh[x - h]^2 - 1) y

Then when

DSolve[lhs[y[x]] == 0, y, x]

doesn't yield an answer, but you know the solution 1/(Cosh[2(x-h)]
+1)^(1/2)  you can use the variation of parameters method

DSolve[lhs[1/(Cosh[2 (x - h)] + 1)^(1/2) w[x]] == 0, w, x]

which yields

{{w -> Function[{x},
C[2] + C[1] (1/2 (-h + x) - 1/4 Sinh[2 (h - x)])]}}

C[2]->1, C[1]->0 corresponds to the solution you already knew, and
C[2]->0, C[1]->1 yields a second, linearly independent solution
y[x_]=1/(Cosh[2(x-h)]+1)^(1/2) (1/2 (-h + x) - 1/4 Sinh[2 (h - x)])

Steve

```

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