How to get a solution involving a symbolic variable with ode45?

I'm trying to work out the maximum horizontal distance that a projectile with air resistance can reach, and the associated launch angle. I currently have a working ODE system that can give the trajectory for the projectile with a given angle using ode45(), that I also have set to stop once it hits the ground again. What I'm having trouble with is getting ode45() to still give me a solution when using a symbolic variable for the angle.

Here is the working code for the projectile:

options = odeset('event',@pjtevent)
pjtode = @(t,y) [y(3); y(4); odefunction([y(3),y(4)])];
v = 100;
theta = pi/4;
[t,y] = ode45(pjtode, [0 25], [0 0 v*cos(theta) v*sin(theta)], options)

pjtevent.m just contains the code to stop the system at the event of it hitting the ground, and odefunction.m is shown below:

function dy = odefunction(x)
K = 0.001;
g = 9.8;
m = 1;
dy = zeros(2,1);
dy(1) = (-K*x(1)^2)/m;
dy(2) = ((-K*x(2)^2)/m)-g;

I thought just setting theta as a symbolic variable would give me a solution depending on theta that I could then optimize, and so tried this:

options = odeset('event',@pjtevent)
pjtode = @(t,y) [y(3); y(4); odefunction([y(3),y(4)])];
v = 100;
syms theta;
[t,y] = ode45(pjtode, [0 25], [0 0 v*cos(theta) v*sin(theta)], options)

However, this returns the following errors:

The following error occurred converting from sym to double:
Error using symengine (line 59)
DOUBLE cannot convert the input expression into a double array.
If the input expression contains a symbolic variable, use VPA.

Error in odefunction (line 6)
dy(1) = (-K*x(1)^2)/m;

Error in @(t,y)[y(3);y(4);odefunction([y(3),y(4)])]

Error in odearguments (line 87)
f0 = feval(ode,t0,y0,args{:});   % ODE15I sets args{1} to yp0.

Error in ode45 (line 113)
[neq, tspan, ntspan, next, t0, tfinal, tdir, y0, f0, odeArgs, odeFcn, ...

Any ideas for how to sort this out as I'm having trouble finding how to get it to work? Thanks.

[Rena Berman]

(Answers Dev) Restored edit

Answer 1

[Steven Lord]

The ode45 function solves ODEs numerically and will not work with symbolic variables. Take a look at this page from the documentation (note: this is the documentation for the most recent release; if you're using an older release, find the equivalent page in your local documentation) for tools and examples for solving ODEs symbolically.

[Alan Weiss]

In addition to what Steve wrote, you might find this example from Global Optimization Toolbox to be relevant.

Alan Weiss

MATLAB mathematical toolbox documentation