       Re: Why doesn't DSolve find all the solutions of this pde

• To: mathgroup at smc.vnet.net
• Subject: [mg72468] Re: [mg72459] Why doesn't DSolve find all the solutions of this pde
• From: Devendra Kapadia <dkapadia at wolfram.com>
• Date: Sat, 30 Dec 2006 05:31:42 -0500 (EST)
• References: <200612291011.FAA29956@smc.vnet.net>

```On Fri, 29 Dec 2006, Valeri Astanoff wrote:

> Good day,
>
> I wonder why DSolve doesn't find all the solutions
> of this pde :
>
> In:=eq = r D[f[r,th],r] - Tan[2th] D[f[r,th],th] == 0;
>
> In:=DSolve[eq ,f[r,th],{r,th}]
>
>> From In:=
> Solve::ifun Inverse functions are being used by Solve,
> so some solutions may not be found; use Reduce
> for complete solution information.
>
> Out={{f[r, th] -> C[Log[(-r)*Sqrt[Sin[2*th]]]]},
> {f[r, th] -> C[Log[r*Sqrt[Sin[2*th]]]]}}
>
> In:=g[r_,th_] = r^2*Sin[2th];
>
> In:=eq/.f -> g
> Out=True
>
> In:=\$Version
> Out=5.1 for Microsoft Windows (January 28, 2005)
>
>
> With my early thanks for any advice
>
>
> V.Astanoff
>
Hello Valeri,

For linear first-order partial differential equations such as the
one considered by you above, DSolve returns a general solution
containing an arbitrary function C. Particular solutions
of the equation can be obtained by specifying the expression
for C.

Thus, the solution r^2*Sin[2th] requested by you can be
found by setting C[x_]:=E^(2x) as shown below (see In).

============================

In:= \$Version

Out= 5.2 for Microsoft Windows (June 20, 2005)

In:= eq = r D[f[r, th], r] - Tan[2 th] D[f[r, th], th] == 0;

In:= (sol = DSolve[eq, f, {r, th}]) // InputForm

Solve::ifun: Inverse functions are being used by Solve, so some solutions
may not be found;
use Reduce for complete solution information.

Solve::ifun: Inverse functions are being used by Solve, so some solutions
may not be found; use Reduce for complete solution information.

Out//InputForm=
{{f -> Function[{r, th}, C[Log[-(r*Sqrt[Sin[2*th]])]]]},
{f -> Function[{r, th}, C[Log[r*Sqrt[Sin[2*th]]]]]}}

In:= (f[r, th] /. sol[]) /. {C[x_] :> E^(2 x)}

2
Out= r  Sin[2 th]

============================

I hope that this helps in answering your question.

Sincerely,

Devendra Kapadia,
Wolfram Research, Inc.

```

• Prev by Date: programming problem about elements taken
• Next by Date: Re: GraphPlot display anomaly
• Previous by thread: Why doesn't DSolve find all the solutions of this pde
• Next by thread: A problem in calculus