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Re: How to get the best expression of an ode's solution

  • To: mathgroup at smc.vnet.net
  • Subject: [mg93639] Re: How to get the best expression of an ode's solution
  • From: dimitris <dimmechan at yahoo.com>
  • Date: Wed, 19 Nov 2008 05:40:26 -0500 (EST)
  • References: <gfuc0q$ei9$1@smc.vnet.net>

On 18 =CD=EF=DD, 14:21, catny <d... at shu.edu.cn> wrote:
> Hello everyone
>
> Here is a linear ODE
>  a f''[x]+f'[x]==h'[x]
> with f[0]=0, f[1]=1
> It is easy to solve it by Mathematica, but the expression is complex. How=
 to get it's simplest form?
>
> Thanks

I don't see a simpler form unless you specify the function h[x].
On my 5.2 version I get

In[3]:=
g=DSolve[{a*Derivative[2][f][x] + Derivative[1][f][x] == Derivative[1=
]
[h][x], f[0] == 0, f[1] == 1}, f[x], x]

Out[3]=
{
  {f[x] -> 1 + Integrate[(E^(1/a - K$1739/a)*(1 + Integrate[Integrate
[(E^(K$1718/a)*Derivative[1][h][K$1718])/a,
             {K$1718, 1, K$1739}]/E^(K$1739/a), {K$1739, 1, 0}]))/(a*
(-1 + E^(1/a))) +
       Integrate[(E^(K$1718/a)*Derivative[1][h][K$1718])/a, {K$1718,
1, K$1739}]/E^(K$1739/a), {K$1739, 1, x}]}}

 K$ variables are used as dummy integration variables.

Perhaps something like the following might help you

In[8]:=
g /. {K$3425 -> K[1], K$3404 -> K[2]}

Out[8]=
{
  {f[x] -> 1 + Integrate[(E^(1/a - K[1]/a)*(1 + Integrate[Integrate[(E^
(K[2]/a)*Derivative[1][h][K[2]])/a, {K[2], 1, K[1]}]/
            E^(K[1]/a), {K[1], 1, 0}]))/(a*(-1 + E^(1/a))) + Integrate
[(E^(K[2]/a)*Derivative[1][h][K[2]])/a, {K[2], 1, K[1]}]/
        E^(K[1]/a), {K[1], 1, x}]}}

Regards
Dimitris Anagnostou


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