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[1]:=eq = r D[f[r,th],r] - Tan[2th] D[f[r,th],th] == 0; > > In[2]:=DSolve[eq ,f[r,th],{r,th}] > >> From In[2]:= > Solve::ifun Inverse functions are being used by Solve, > so some solutions may not be found; use Reduce > for complete solution information. > > Out[2]={{f[r, th] -> C[1][Log[(-r)*Sqrt[Sin[2*th]]]]}, > {f[r, th] -> C[1][Log[r*Sqrt[Sin[2*th]]]]}} > > In[3]:=g[r_,th_] = r^2*Sin[2th]; > > In[4]:=eq/.f -> g > Out[4]=True > > In[5]:=$Version > Out[5]=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[1]. Particular solutions of the equation can be obtained by specifying the expression for C[1]. Thus, the solution r^2*Sin[2th] requested by you can be found by setting C[1][x_]:=E^(2x) as shown below (see In[4]). ============================ In[1]:= $Version Out[1]= 5.2 for Microsoft Windows (June 20, 2005) In[2]:= eq = r D[f[r, th], r] - Tan[2 th] D[f[r, th], th] == 0; In[3]:= (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[3]//InputForm= {{f -> Function[{r, th}, C[1][Log[-(r*Sqrt[Sin[2*th]])]]]}, {f -> Function[{r, th}, C[1][Log[r*Sqrt[Sin[2*th]]]]]}} In[4]:= (f[r, th] /. sol[[1]]) /. {C[1][x_] :> E^(2 x)} 2 Out[4]= r Sin[2 th] ============================ I hope that this helps in answering your question. Sincerely, Devendra Kapadia, Wolfram Research, Inc.

**References**:**Why doesn't DSolve find all the solutions of this pde***From:*"Valeri Astanoff" <astanoff@gmail.com>