Solve complicated second order differential equation

[Miguel_s]

Hello everyone,

 

I have been having a hard time with this equation...

 

I would really like to know if anybody has an idea on how to solve it...

 

All the suggestions are very much appreciated :)

 

[studiot]

It would be a lot easier to understand and be sure of your notation if you provided a symbols key.

is FcH+c meant to be  (F) (cH+)c  ?

Have you plotted a direction or characteristic field for the equation?

[mathematic]

It loos like an ordinary (not partial) diff. eq. in r.  Which symbols represent functions of r?

[Miguel_s]

The following 6 symbols below are constants:

  ...  ...  ... F... R... T...

And what I need is to solve the equation for , which varies with r.

 

Regarding your questions, I haven't plotted the characteristic field as I never heard about it...

Only  is a function of r...

 

[studiot]

4 hours ago, Miguel_s said:

The following 6 symbols below are constants:

  ...  ...  ... F... R... T...

And what I need is to solve the equation for , which varies with r.

 

Regarding your questions, I haven't plotted the characteristic field as I never heard about it...

Only  is a function of r... 

 

Since you only have two variables I agree with mathematic that this is an ODE.

I would proceed as follows:

Replacing all your constants and collecting them together, you only need 3 constants.

I have started with four, A, B, C and D and finally added E = B/A

[math] - \frac{d}{{dr}}\left( {\varepsilon r\frac{{d\varphi }}{{dr}}} \right) = B{e^{ - \left( {C\varphi  + D} \right)}}[/math]

[math] - \varepsilon \frac{d}{{dr}}\left( {r\frac{{d\varphi }}{{dr}}} \right) = B{e^{ - \left( {C\varphi  + D} \right)}}[/math]

[math]A\frac{d}{{dr}}\left( {r\frac{{d\varphi }}{{dr}}} \right) = B{e^{ - \left( {C\varphi  + D} \right)}}[/math]

[math]A\left( {\frac{{d\varphi }}{{dr}} + r\frac{{{d^2}\varphi }}{{d{r^2}}}} \right) = B{e^{ - \left( {C\varphi  + D} \right)}}[/math]

[math]\left( {\frac{{d\varphi }}{{dr}} + r\frac{{{d^2}\varphi }}{{d{r^2}}}} \right) = E{e^{ - \left( {C\varphi  + D} \right)}}[/math]

 

 

[Miguel_s]

17 minutes ago, studiot said:

Since you only have two variables I agree with mathematic that this is an ODE.

I would proceed as follows:

Replacing all your constants and collecting them together, you only need 3 constants.

I have started with four, A, B, C and D and finally added E = B/A

−ddr(εrdφdr)=Be−(Cφ+D)

−εddr(rdφdr)=Be−(Cφ+D)

Addr(rdφdr)=Be−(Cφ+D)

A(dφdr+rd2φdr2)=Be−(Cφ+D)

(dφdr+rd2φdr2)=Ee−(Cφ+D)

 

 

It definitely looks better. From that stage on is where I can't just go through... I have tried variable substitution before and nothing...

 

If you have any idea I would really appreciate