Solving nonlinear coupled differential equations in Mathematica 7.0

[SK]

Here is a set of equations I would like to solve

y''[x]==y+z'[x]
z[x]^3=y'[x]+3

With boundary conditions of y[1]==2 and y[0]==3

Using NDSolve on these equations, Mathematica says the order of the
equations is 3 and it has only 2 initial conditions. But the order of
this system of equations is 2 as far as I see (since order is defined
as the highest derivative)

When I do try to put another boundary condition in like z[0]==0
Mathematica spits out that it cant solve for the derivatives and is
using a mass matrix method (error:ntdvmm) and then it says that it has
significant errors (error:berr) and will return the best solution
found.

Any help on this matter will be greatly appreciated.
Thanks

[Jens-Peer Kuska]

Hi,

your equations are not a differential equation for
z[x] because

ode = {y''[x] == y[x] + z'[x],
z[x]^3 == y'[x] + 3};

ode /. Solve[D[#, x] & /@
Last[ode], z'[x]][[1]]

gives:

{Derivative[2][y][x] == y[x] + Derivative[2][y][x]/(3*z[x]^2),
z[x]^3 == 3 + Derivative[1][y][x]}

that does not include z'[x] any more and more over
z[x] is complete undetermined.

Setting

ode1 = ode /. Solve[D[#, x] & /@
Last[ode], z'[x]][[1]];

yodes= First /@ (ode1 /. Solve[Last[ode1], z[x]])

gives three possible equations for y[x] and

NDSolve[
{#, y[1] == 2 , y[0] == 3},
y[x], {x, 0, 1}] & /@ yodes

will solve it with some numerical error messages
because I'm not able to to take the right solution
for z[x]

Regards
Jens

[Torsten Hennig]

> Here is a set of equations I would like to solve
>
> y''[x]==y+z'[x]
> z[x]^3=y'[x]+3
>
> With boundary conditions of y[1]==2 and y[0]==3
>
> Using NDSolve on these equations, Mathematica says
> the order of the
> equations is 3 and it has only 2 initial conditions.
> But the order of
> this system of equations is 2 as far as I see (since
> order is defined
> as the highest derivative)
>
> When I do try to put another boundary condition in
> like z[0]==0
> Mathematica spits out that it cant solve for the
> derivatives and is
> using a mass matrix method (error:ntdvmm) and then it
> says that it has
> significant errors (error:berr) and will return the
> best solution
> found.
>
> Any help on this matter will be greatly appreciated.
> Thanks
>

z^3 = y' + 3 -> 3*z^2*z' = y''
Inserting in the first equation gives
3*z^2*z' = y + z'
or
z'=y/(3*z^2-1)

So you should solve the following two equations

(I) z'=y/(3*z^2-1)
(II) y'=z^3-3

with appropriate boundary conditions.

Best wishes
Torsten.