Solve third order, non-linear, ODE with indefinite BC's?

[Topher925]

I have a third order, non-linear, homogeneous, constant coefficient ODE that I need to solve but have no idea how to do it. To make matters worse, one of the boundary conditions are indefinite. Here's the equation,

y''' + y*y'' - y'^2 + 1 = 0

and the BC's
y(0) = 0
y'(0) = 0
y'($$\infty$$) = 1

Does this equation have a general or analytical (non numerical) solution? I have access to Maple, just not at the moment, and would like to try and solve it analytically without computer first. Any suggestions on how to solve it? Can it be solved?

 

[jackmell]

I have a third order, non-linear, homogeneous, constant coefficient ODE that I need to solve but have no idea how to do it. To make matters worse, one of the boundary conditions are indefinite. Here's the equation,

y''' + y*y'' - y'^2 + 1 = 0

and the BC's
y(0) = 0
y'(0) = 0
y'($$\infty$$) = 1

Does this equation have a general or analytical (non numerical) solution? I have access to Maple, just not at the moment, and would like to try and solve it analytically without computer first. Any suggestions on how to solve it? Can it be solved?

It's called the Falkner-Skan equation but maybe you know that already. Maybe a Google can help you. Wikipedia says it can't be solve analytically. Here's my attempt to solve it numerically in a limited range, say (0,10) in Mathematica:

myend = 10;
sols = Map[
   First[NDSolve[{y'''[t] + y[t] y''[t] - (y'[t])^2 + 1 == 0, 
       y[0] == y'[0] == 0, y'[myend] == 1}, y, t, 
      Method -> {"Shooting", 
        "StartingInitialConditions" -> {y[0] == 0, y'[0] == 0, 
          y''[0] == #}}]] &, {1.7, 1.75, 1.8}];
Plot[Evaluate[y[t] /. sols], {t, 0, myend}, 
 PlotStyle -> {Black, Blue, Green}, AspectRatio -> 4/3]

And that code returns $y''(0)\approx 1.233$

 

[yus310]

see attached... it solves falkner skan in maple..

any questions, fire away