# How to solve two different ode system which is related?

I need help on solving two systems which are related to each other. Basically, first I have to solve a non-linear ODE system. Then, I have to solve another system of equations, I have to ask mathematica to get the values from first system, in order to solve the second system.

The first system is this one:

inf = 50;
s = ParametricNDSolve[{H'[η] == -2*F[η] - ((1 - n)/(1 + n))*η*F'[η],
F[η]^2 - (G[η] + 1)^2 + (H[η] + ((1 - n)/(1 + n))*η*F[η])*F'[η] ==
(F'[η]^2 + G'[η]^2)^((n - 1)/2)*F''[η] + F'[η]*((n - 1)*(F'[η]^2 + G'[η]^2)^((n - 3)/2)
*(F'[η]*F''[η] + G'[η]*G''[η])),
2*F[η]*(G[η] + 1) + (H[η] + ((1 - n)/(1 + n))*η*F[η])*G'[η] ==
(F'[η]^2 + G'[η]^2)^((n - 1)/2)*G''[η] + G'[η]*((n - 1)*(F'[η]^2 + G'[η]^2)^((n - 3)/2)
*(F'[η]*F''[η] + G'[η]*G''[η])),
F[0] == 0, G[0] == 0, H[0] == 0,
F'[inf] == H[inf] F[inf], G'[inf] == H[inf] (G[inf] + 1)},
{F, G, H, F', G'}, {η, 0, inf}, {n, Fp0, Gp0},
Method -> {"Shooting", "StartingInitialConditions" ->
{F[0] == 0, G[0] == 0, H[0] == 0, F'[0] == Fp0, G'[0] == Gp0}}];


The value of F[η], G[η] and H[η] that I get from the first system will be used to solve the second system.

I believe there is a lot of mistakes i've done to solve the second question. Hence, I need help. thank you :)

This is the second system:

inf = 20;
alphabar=1;
betabar=1;
omegabar=0;
R=1;
sol =First@ NDSolve[{y1'[\[Eta]] == y2[\[Eta]] ,
y2'[\[Eta]]==(1/((F'[\[Eta]]^2 + G'[\[Eta]]^2)^((n - 1)/2)))*(H[\[Eta]]-((n - 1)*(F'[\[Eta]]^2 + G'[\[Eta]]^2)^((n - 3)/2)*(F'[\[Eta]]*F''[\[Eta]] +
G'[\[Eta]]*G''[\[Eta]])))*y2[\[Eta]]+(1/((F'[\[Eta]]^2 + G'[\[Eta]]^2)^((n - 1)/2)))*(\[ImaginaryI]*R*(alphabar*F[\[Eta]]+betabar*G[\[Eta]]-omegabar)+F[\[Eta]])*y1[\[Eta]]-(2/((F'[\[Eta]]^2 + G'[\[Eta]]^2)^((n - 1)/2)))*(G[\[Eta]]+1)*y5[\[Eta]]+(R/((F'[\[Eta]]^2 + G'[\[Eta]]^2)^((n - 1)/2)))*F'[\[Eta]]*y3[\[Eta]]+\[ImaginaryI]*(R/((F'[\[Eta]]^2 + G'[\[Eta]]^2)^((n - 1)/2)))*alphabar*y4[\[Eta]],
y3'[\[Eta]]==-(\[ImaginaryI]*alphabar*(1/R))*y1[\[Eta]]-\[ImaginaryI]*betabar*y5[\[Eta]],
y4'[\[Eta]]==-((\[ImaginaryI]*R*(alphabar*F[\[Eta]]+betabar*G[\[Eta]]-omegabar)+H'[\[Eta]])/R)*y3[\[Eta]]-(H[\[Eta]]/R)*(-\[ImaginaryI]((alphabar*(1/R))*y1[\[Eta]]-\[ImaginaryI]*betabar*y5[\[Eta]]))-(1/R)*(((F'[\[Eta]]^2 + G'[\[Eta]]^2)^((n - 1)/2))*y3''[\[Eta]]+y3'[\[Eta]]*(n - 1)((F'[\[Eta]]^2 + G'[\[Eta]]^2)^((n - 3)/2))(F'[\[Eta]]* F''[\[Eta]]+ G'[\[Eta]]*G''[\[Eta]])),
y5'[\[Eta]]==y6[\[Eta]],
y6'[\[Eta]]==(1/((F'[\[Eta]]^2 + G'[\[Eta]]^2)^((n - 1)/2)))*(H[\[Eta]]-((n - 1)*(F'[\[Eta]]^2 + G'[\[Eta]]^2)^((n - 3)/2)*(F'[\[Eta]]*F''[\[Eta]] +
G'[\[Eta]]*G''[\[Eta]])))*y6[\[Eta]]+(1/((F'[\[Eta]]^2 + G'[\[Eta]]^2)^((n - 1)/2)))*(\[ImaginaryI]*R*(alphabar*F[\[Eta]]+betabar*G[\[Eta]]-omegabar)+F[\[Eta]])*y5[\[Eta]]-(2/((F'[\[Eta]]^2 + G'[\[Eta]]^2)^((n - 1)/2)))*(G[\[Eta]]+1)*y1[\[Eta]]+(R/((F'[\[Eta]]^2 + G'[\[Eta]]^2)^((n - 1)/2)))*F'[\[Eta]]*y3[\[Eta]]+\[ImaginaryI]*(R/((F'[\[Eta]]^2 + G'[\[Eta]]^2)^((n - 1)/2)))*betabar*y4[\[Eta]],  y1[0] == 0, y2[0] == 0, y3[0] == 0, y4[0] == 0, y5[0] == 0, y6[0] == 0,y1[inf] == 0, y2[inf] == 0, y3[inf] == 0,y4[inf] == 0, y5[inf] == 0, y6[inf] == 0} /. n -> 1.0,
{y1,y2,y3,y4,y5,y6}, {\[Eta], 0.00001, inf}];


This is the error:

 NDSolve
:Derivative order  1.  in term  y1
(1.)
[η]  should be a non-negative machine-sized integer.


Thank you so much if anyone can help me on this.

Attached is the system:

I replaced η with y, U with F, V with G and W with H.

• The first part of this question is addressed here. Feb 3, 2017 at 9:27
• Because all but the third ODE in the second block of code are first order, there should be only seven boundary conditions, not twelve. Please specify which you wish to use. Also, the fourth ODE contains y3''[η] as well as y4'[η]. Is this correct? Feb 3, 2017 at 10:23
• Also, since these equations are linear and homogeneous in y the solution is zero. Feb 3, 2017 at 11:23
• Why don't you try to solve the two systems as a one system?
– zhk
Feb 3, 2017 at 12:50
• What is i in the 2nd system?
– zhk
Feb 3, 2017 at 14:26

Here I will illustrate with a simple example to solve two couple systems of odes.

Method I

(*The first system*)
Ode1 = x'[t] == 1/y[t];
Ode2 = y'[t] == x[t];
ics := {x[0] == 1, y[0] == 2};
sys = Join[{Ode1, Ode2}, ics];
soln = First@NDSolve[sys, {x, y}, {t, 0, 10}];
x1[t_] := Evaluate[x[t] /. soln]; (*assign the interpolating function x[t] to x1[t]*)
y1[t_] := Evaluate[y[t] /. soln]; (*assign the interpolating function y[t] to y1[t]*)
(*The second system*)
Ode3 = x2'[t] == x1[t];
Ode4 = y2'[t] == x2[t] + y1[t] + y2[t];
ics2 := {x2[0] == -1, y2[0] == -2};
sys1 = Join[{Ode3, Ode4}, ics2];
soln1 = First@NDSolve[sys1, {x2, y2}, {t, 0, 10}];
Plot[{x1[t], y1[t], {x2[t], y2[t]} /. soln1}, {t, 0, 10}]


Method II

Solve the two systems as one,

Ode3 = x2'[t] == x[t];
Ode4 = y2'[t] == x2[t] + y[t] + y2[t];
combsys12 = Join[{Ode1, Ode2, Ode3, Ode4}, ics, ics2];
combsoln = First@NDSolve[combsys12, {x, y, x2, y2}, {t, 0, 10}];
Plot[{x[t], y[t], x2[t], y2[t]} /. combsoln, {t, 0, 10}]