Solving a Linear ODE - Help!

I have an ode

(2*A*k + 3*G*h(t)^(1/2))*diff(h(t), t) == 2*G*h(t)*(h(t)^(1/2) + (A*k)/G)

that I need to plot as h(t) vs t. ODE45 (and ODE23) returns:

 Undefined function 'exist' for input arguments of type 'sym'.
Error in odearguments (line 34)
    if exist(ode)==2 && ( nargout(ode)<3 && nargout(ode)~=-1 )
Error in ode45 (line 115)
    odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);

Solve returns an empty struct, and dSolve returns an implicit answer

ans =

                                                    0
   root(2*G*z^3 + 2*A*k*z^2 + exp(C24/2 + t), z, 1)^2

or if I raise the MaxDegree to three, six very complex answers with several constants that I don't know how to find. Does anyone know another simple method to solve this ode? Thank you in advance!

Answer 1

[Star Strider]

Try this:

syms A G k h(t) Y 

Eqn = (2*A*k + 3*G*h(t)^(1/2))*diff(h(t), t) == 2*G*h(t)*(h(t)^(1/2) + (A*k)/G)

[ODEF, Subs] = odeToVectorField(Eqn);

DE = matlabFunction(ODEF, 'Vars',{t,Y,[A,G,k]});

A = 0.3;                                                % Provide Correct Values
G = 0.5;                                                % Provide Correct Values
k = 0.7;                                                % Provide Correct Values
h0 = 0.1;                                               % Provide Correct Values
tspan = [0 1];                                          % Provide Correct Values
[t,y] = ode45(@(t,Y)DE(t,Y,[A,G,k]), tspan, h0);

figure(1)
plot(t, y)
grid

[Rosie Morris]

Thank you, that worked perfectly for that equation :) how would I go about it if I had a third order ode? Could I use reduceDifferentialOrder(eqn0, h(t)) then run through the same process?

Thanks for your help!

[Star Strider]

As always, my pleasure!

‘Could I use reduceDifferentialOrder(eqn0, h(t)) then run through the same process?’

Try it! I have not used reduceDifferentialOrder and odeFunction because I have not needed them. The procedure I use here has worked for me. Note that reduceDifferentialOrder apparently requires odeFunction, so the procedure I used here (specifically matlabFunction) may not work with it. I would use the procedure I use here with your third-order ODE. If it works, go with it, otherwise experiment with reduceDifferentialOrder and odeFunction.

[Karan Gill]

It should work. Follow the doc here: https://www.mathworks.com/help/symbolic/examples/solve-a-second-order-differential-equation-numerically.html

Answer 2

[Rosie Morris]

I've tried both methods (odeToVectorField/matlabFunction and reduceDifferentialOrder/odeFunction) but I get stuck on the ode solver with the error:

      Index exceeds matrix dimensions.

      Error in
      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.*in3(:,12).^2.*in3(:,13).*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*Y(2).*3.0-in3(:,1).^2.*in3(:,8).^2.*in3(:,9).*in3(:,10).^2.*in3(:,2).*in3(:,11).*in3(:,12).*in3(:,13).^2.*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*Y(2).*3.0-in3(:,1).^2.*in3(:,8).^2.*in3(:,9).*in3(:,10).^2.*in3(:,2).*in3(:,11).^2.*in3(:,12).*in3(:,13).*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*Y(2).*3.0-in3(:,1).^2.*in3(:,8).^2.*in3(:,9).^2.*in3(:,10).*in3(:,2).*in3(:,11).*in3(:,12).^2.*in3(:,13).*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*Y(2).*3.0-in3(:,1).^2.*in3(:,8).^2.*in3(:,9).^2.*in3(:,10).*in3(:,2).*in3(:,11).^2.*in3(:,12).*in3(:,13).*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*Y(2).*3.0-in3(:,1).*in3(:,4).*in3(:,8).*in3(:,9).^2.*in3(:,10).^2.*in3(:,2).*in3(:,11).*in3(:,12).*in3(:,13).^2.*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*3.0-in3(:,1).*in3(:,4).*in3(:,8).*in3(:,9).^2.*in3(:,10).^2.*in3(:,2).*in3(:,11).*in3(:,12).^2.*in3(:,13).*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*3.0-in3(:,1).*in3(:,4).*in3(:,8).*in3(:,9).^2.*in3(:,10).^2.*in3(:,2).*in3(:,11).^2.*in3(:,12).*in3(:,13).*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*6.0-in3(:,1).*in3(:,4).*in3(:,8).^2.*in3(:,9).*in3(:,10).^2.*in3(:,2).*in3(:,11).*in3(:,12).*in3(:,13).^2.*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*3.0-in3(:,1).*in3(:,4).*in3(:,8).^2.*in3(:,9).*in3(:,10).^2.*in3(:,2).*in3(:,11).*in3(:,12).^2.*in3(:,13).*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*6.0-in3(:,1).*in3(:,4).*in3(:,8).^2.*in3(:,9).*in3(:,10).^2.*in3(:,2).*in3(:,11).^2.*in3(:,12).*in3(:,13).*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*3.0-in3(:,1).*in3(:,4).*in3(:,8).^2.*in3(:,9).^2.*in3(:,10).*in3(:,2).*in3(:,11).*in3(:,12).*in3(:,13).^2.*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*6.0-in3(:,1).*in3(:,4).*in3(:,8).^2.*in3(:,9).^2.*in3(:,10).*in3(:,2).*in3(:,11).*in3(:,12).^2.*in3(:,13).*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*3.0-in3(:,1).*in3(:,4).*in3(:,8).^2.*in3(:,9).^2.*in3(:,10).*in3(:,2).*in3(:,11).^2.*in3(:,12).*in3(:,13).*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*3.0-in3(:,1).^2.*sqrt(in3(:,6)).*in3(:,8).^2.*in3(:,9).^2.*in3(:,10).^2.*in3(:,11).^2.*in3(:,12).^2.*in3(:,13).^2.*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).*Y(2).*Y(3).*1.8e1-in3(:,1).*in3(:,8).^2.*in3(:,9).^2.*in3(:,10).^2.*in3(:,2).*in3(:,11).*in3(:,12).^2.*in3(:,13).^2.*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*Y(3).*3.0-in3(:,1).*in3(:,8).^2.*in3(:,9).^2.*in3(:,10).^2.*in3(:,2).*in3(:,11).^2.*in3(:,12).*in3(:,13).^2.*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*Y(3).*3.0-in3(:,1).*in3(:,8).^2.*in3(:,9).^2.*in3(:,10).^2.*in3(:,2).*in3(:,11).^2.*in3(:,12).^2.*in3(:,13).*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*Y(3).*3.0+in3(:,1).^2.*in3(:,8).*in3(:,9).^2.*in3(:,10).^2.*in3(:,2).*in3(:,11).*in3(:,12).^2.*in3(:,13).^2.*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*Y(3).*3.0+in3(:,1).^2.*in3(:,8).^2.*in3(:,9).*in3(:,10).^2.*in3(:,2).*in3(:,11).^2.*in3(:,12).*in3(:,13).^2.*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*Y(3).*3.0+in3(:,1).^2.*in3(:,8).^2.*in3(:,9).^2.*in3(:,10).*in3(:,2).*in3(:,11).^2.*in3(:,12).^2.*in3(:,13).*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*Y(3).*3.0+in3(:,4).*in3(:,8).^2.*in3(:,9).^2.*in3(:,10).^2.*in3(:,2).*t.*in3(:,11).*in3(:,12).*in3(:,13).*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*3.0+in3(:,1).*in3(:,4).*in3(:,8).*in3(:,9).^2.*in3(:,10).^2.*in3(:,2).*t.*in3(:,11).*in3(:,12).*in3(:,13).*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*3.0+in3(:,1).*in3(:,4).*in3(:,8).^2.*in3(:,9).*in3(:,10).^2.*in3(:,2).*t.*in3(:,11).*in3(:,12).*in3(:,13).*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*3.0+in3(:,1).*in3(:,4).*in3(:,8).^2.*in3(:,9).^2.*in3(:,10).*in3(:,2).*t.*in3(:,11).*in3(:,12).*in3(:,13).*in3(:,3).*exp(t./in3(:,11)).*exp(t./in3(:,12)).*exp(t./in3(:,13)).*Y(1).^(3.0./2.0).*3.0).*(1.0./3.0))./(in3(:,13).*(in3(:,1).*in3(:,8).^2.*in3(:,9).^2.*in3(:,10).^2.*in3(:,2).*in3(:,11).^2.*in3(:,12).^2.*in3(:,13)+in3(:,1).^2.*sqrt(in3(:,6)).*in3(:,8).^2.*in3(:,9).^2.*in3(:,10).^2.*in3(:,11).^2.*in3(:,12).^2.*in3(:,13).*sqrt(Y(1)).*4.0-in3(:,1).*in3(:,8).^2.*in3(:,9).^2.*in3(:,10).^2.*in3(:,2).*in3(:,11).^2.*in3(:,12).^2.*in3(:,13).*in3(:,3)))]

      Error in @(t,Y)DE(t,Y,[G,k,v,V,G,R,y,g1,g2,g3,tor1,tor2,tor3,h0,omega])

      Error in odearguments (line 90)
      f0 = feval(ode,t0,y0,args{:});   % ODE15I sets args{1} to yp0.

      Error in ode45 (line 115)
          odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);

>> (my equation is REALLY long :( )

Thanks for your help guys!

[Star Strider]

The vector (third argument) in ‘DE’ has 15 elements. The highest index I was able to find for ‘in3’ is 14 (including using Firefox Search on this page).

[Rosie Morris]

I reduced the number of variables by adding in their values earlier and it worked!!! Thank you so much Star Strider! I've been stuck on this for so long :3

[Star Strider]

As always, my pleasure!