       Re: Is it possible to solve this differential equation?

• To: mathgroup at smc.vnet.net
• Subject: [mg89028] Re: Is it possible to solve this differential equation?
• From: Roland Franzius <roland.franzius at uos.de>
• Date: Sat, 24 May 2008 03:52:42 -0400 (EDT)
• References: <g15qst\$pgd\$1@smc.vnet.net>

```Wei Wang schrieb:
> Hi,
>
> Is it possible to solve the following equation?
>
> DSolve[A (Sqrt[B/y[x]] - 1) (1 - Exp[-y[x]/B]^(1/2) == D[y[x], x], =
> y[x], x]

If I read correctly what your equation is something like

((dy/B) / (A dx) ) == (Sqrt[B/y] - 1) (1 - Exp[-y/B]^(1/2)

so with y/B = z, u = A x you have

du = dz / (Sqrt[1/z] - 1) * (1 - Exp[-z]^(-1/2)

Integrate[dz / (Sqrt[1/z] - 1) * (1 - Exp[-z]^(-1/2),z]

gives

u==(Sqrt[2*Pi] + 2*E^(z/2)*(Sqrt[E^(-z)] + Sqrt[1/z])*Sqrt[-z] -
Sqrt[2*Pi]*Erf[Sqrt[-z]/Sqrt])/
(E^(z/2)*(Sqrt[E^(-z)]*Sqrt[1/z]*Sqrt[-z]))

This transcendental equation is solvable for u->z[u] only numerical.

Of course, if you are interested in numerical solutions you can NDSolve
the original equation directly, too.

--

Roland Franzius

```

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