Matlab gives no result when I use the ODE (Ordinary Differential Equations)

When trying to find the symbolic solution in the following code, I got a warning and empty sym!

syms y(t) y0 Dy0
A=1;
B=2;
Dy = diff(y,t);
D2y = diff(y,t,2);
ode = diff(y,t,2)-A*(y)^2-B^2*y==0;
cond1 = y(0) == 1;
cond2 = Dy(1) == 0;
conds = [cond1 cond2];
ySol(t) = dsolve(ode,conds);
ySol = simplify(ySol, 'Steps',20)
disp(ySol(t))

Warning: Explicit solution could not be found.

> In dsolve (line 201)

In symbolic_Fun (line 11)

ySol(t) =

[ empty sym ]

[ empty sym ]

What was my mistake?

Please advise.

Answer 1

[Star Strider]

The differential equation is nonlinear (the ‘y^2’ term) and very few nonlinear differential equations have analytic solutions. You need to integrate it numerically.

Try this:

syms y(t) y0 Dy0 T Y 
A=1;
B=2;
Dy = diff(y,t);
D2y = diff(y,t,2);
ode = diff(y,t,2)-A*(y)^2-B^2*y==0;
cond1 = y(0) == 1;
cond2 = Dy(1) == 0;
conds = [cond1 cond2];
[VF,Subs] = odeToVectorField(ode)
odefcn = matlabFunction(VF, 'Vars',{T,Y})

initconds = [1, 0];                                 % Use The ÑSubs’ Output To Determine These Positions In The Vector
tspan = [0 1];
[t,y] = ode45(odefcn, tspan, initconds);

figure
plot(t, y)
grid
legend(string(Subs))

It goes to +Inf at about t=1.9.

[Samer Atawneh]

Dear Strider,

Thank you very much for your kind reply. Appreciated.

[Star Strider]

My pleasure.

If my Answer helped you solve your problem, please Accept it!

[Samer Atawneh]

Dear Strider,

Thank you very much for your kind reply. Actually, I got the following warining and the execution stopped.

Warning: string is obsolete and will be discontinued. Use char instead.

Pluse, the result should be in a decreasing way not in increasing. Please check what figure draws.

Waiting to hear from you.

Thank you again.

[Star Strider]

The string data type that I use in my code was introduced in R2016b. Since you have an earliler release, substitute this line for the one I originally posted:

legend(sprintfc('%s',Subs))

That worked when I tested it, and it should work for you. The sprintfc function is undocumented (and all releases have it), and works like sprintf except that it returns a cell array as its output.

[Samer Atawneh]

Dear Strider,

Thank you very much for your kind reply. I got the following error:

Error using sprintfc

Function is not defined for 'sym' inputs.

[Star Strider]

Interesting. It worked for me (in R2019b) or I would not have posted it. I have no idea what will work in your MATLAB version, since I do not know which one it is.

In that situation, just do it manually then:

legend('y','Dy')

[Samer Atawneh]

Dear Strider,

Thank you very much. Appreciated.

[Star Strider]

As always, my pleasure!