NDSolve a system of differential equations with initial conditions?

Questions
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1

I don know what is wrong with my initial conditions in the following problem

sol = NDSolve[{x'''[t] == y'[t] x[t], 
   y''[t] == -(1/(x[t]^2 + y[t]^2)), x[0] == 1, x'''[0] == 0, 
   y'[0] == 0, y''[0] == 0}, {x, y}, {t, 0, 100}]

Returns 2 errors:

NDSolve::icordinit: The initial values for all the dependent variables are not explicitly specified. NDSolve will attempt to find consistent initial conditions for all the variables.

and

NDSolve::ndnco: The number of constraints (4) (initial conditions) is not equal to the total differential order of the system plus the number of discrete variables (5).

UPDATE

Sorry… I had made up the above DE to test how NDSolve works ,but use a bad example. Here is another .. please comment

NDSolve[{x'''[t] == 
   x[t] Derivative[1][y][t], (y^\[Prime]\[Prime])[t] == -(1/(
    x[t]^2 + y[t]^2)), x[0] == 1, x'''[0] == 0, y[0] == 1/8, 
  Derivative[1][y][0] == 0, (y^\[Prime]\[Prime])[0] == 1/2}, {x, 
  y}, {t, 0, 1}]

The errors now are :

NDSolve::icordinit: The initial values for all the dependent variables are not explicitly specified. NDSolve will attempt to find consistent initial conditions for all the variables.
NDSolve::icres: NDSolve has computed initial values that give a zero residual for the differential system, but some components are different from those specified. If you need them to be satisfied, giving explicit initial values to all dependent variables is recommended.

2

The condition x'''[0] == 0 is superfluous, because it is a consequence of the first equation and of y'[0] == 0. The condition y''[0] == 0 is not compatible with the second equation.

3

Thank you.. I made up the DE problem . I am looking to test how NDsolve does works. I have made another correction but still don’t know why I am no getting anywhere.

4

Jose,

Gianluca is correct. Your initial conditions should look something like this:

sol = NDSolve[{x'''[t] == y'[t] x[t], 
   y''[t] == -(1/(x[t]^2 + y[t]^2)), x[0] == 1, x''[0] == 1, 
   x'[0] == 1, y'[0] == 0, y[0] == 0}, {x, y}, {t, 0, 100}]

Regards,

Neil

5

Jose,

2 problems with your update. You have a typo in 2 places: y^‘’ should just be y’’

Also, you should not specify initial conditions on x’‘’ because it is determined from an equation. Same for y’'. Both are not valid initial conditions. (see @Gianluca Gorni post above)

This works: (as would any combination of initial conditions that does not involve x’‘’ nor y’')

NDSolve[{x'''[t] == x[t] Derivative[1][y][t], 
  y''[t] == -(1/(x[t]^2 + y[t]^2)), x[0] == 1, x''[0] == 0, 
  y[0] == 1/8, Derivative[1][y][0] == 0, y'[0] == 1/2}, {x, y}, {t, 0,
   1}]

Regards

6

ClearAll["Global`"]
[Nu] = 510^-5;
k = 2*[Pi][Nu];
sol = NDSolve[{[Nu]u’'[y] + k(y)^2Abs[u’[y]]*u’‘[y] == 0,
u[0] == 0, u’[0.999553801] == 0}, u, {y, 0, 0.999553801}]

I cant understand my error it says NDSolve::ndode: The equations {0[0.999554]==0} are not differential equations or initial conditions in the dependent variables {u}.

7

The code

ClearAll["Global`*"]
\[Nu] = 5*10^-5;
k = 2*\[Pi]*\[Nu];
sol = NDSolve[{\[Nu]*u''[y] + k*(y)^2*Abs[u'[y]]*u''[y] == 0, 
   u[0] == 0, u'[0.999553801] == 0}, u, {y, 0, 0.999553801}]

works fine for me. It gives the null function.