converting a second degree differential equation into a first order

I am trying to reduce the order of a function but I always get an error message :

syms y(t);

eqn = diff(y,2) + c*y' + k*y == F0sin(w*t) ;

cond1 = y(0) == 4 ;

cond2 diff(y0) == 0 ;

V = odeToVectorField(eqn);

Answer 1

[Star Strider]

Declare all the variables, and for good measure, return the second output from odeToVectorField because it tells you wnat the function is doing internally, and the substitutions it made.

syms c k F0 w y(t)
D1y = diff(y);
D2y = diff(D1y);
eqn = D2y + c*D1y + k*y == F0*sin(w*t) ;
% cond1 = y(0) == 4 ; 
% cond2 = diff(y0) == 0 ;
[V,Subs] = odeToVectorField(eqn)

V = 

Subs = 

Now, convert this into an anonymous function that the nmumerical ordinary differntial equation solvers can work with, and integrate it to get a numerical result.

.

.

[Anas Gharsa]

I really appreciate it!! thank you !!! But it stiil gives me error!!!

'syms' requires Symbolic Math Toolbox.

Error in finite_differences (line 3)

syms c k F0 w y(t)

[Star Strider]

As always, my pleasure!

Yes, it does.

The original posted code uses Symbolic Math Toolbox syntax and function calls, so I assumed it was available.

In any event, it is still possible to use it here:

syms c k F0 w y(t) t Y 
D1y = diff(y);
D2y = diff(D1y);
eqn = D2y + c*D1y + k*y == F0*sin(w*t) ;
% cond1 = y(0) == 4 ; 
% cond2 = diff(y0) == 0 ;
[V,Subs] = odeToVectorField(eqn)

V = 

Subs = 

Vfcn = matlabFunction(V, 'Vars',{t,Y,c,F0,k,w})

Vfcn = function_handle with value:
    @(t,Y,c,F0,k,w)[Y(2);F0.*sin(t.*w)-c.*Y(2)-k.*Y(1)]

Use that with ode45 (or the function of your choice) to numerically integrate it.

Download and install the Symbolic Math Toolbox as soon as possible to have access to all its functions.

.

[Anas Gharsa]

Thank you so much!!! that was so helpful!! but i have 2 more questions !! i did what u said and still gives me error !!!

Unrecognized function or variable 'y0'.

Error in finite_differences (line 8)

cond2 = diff(y0) == 0 ;

and the last question is : can i use the finite_differences or heun's method to solve it ??

[Star Strider]

As always, my pleasure!

Note — I commented-out the ‘cond1’ and ‘cond2’ assignments because they are not used as symbolic variables in this instance. Instead, assign the values to an initial conditions vector as:

ic = [4; 0];

(this is one instance where the ‘Subs’ result becomes useful) in order to use them with the numeric ODE integration functions.

So long as the method you use integrate it accepts functions such as ‘Vfcn’ in that form, it can be used with them. (I designed it to work with the MATLAB numeric ODE integrators, since I thought that was the eventual objective.)

.

[Anas Gharsa]

Thank you again for your help!!! I really appreciate it

[Star Strider]

As always, my pleasure!