How to solve this ODE

I'm trying to solve the ODE A*(y'') + B*sin(C*y) + D(y') = 0 where y depends on t, y' is dy/dt and y'' is d2y/dt2, and it has the initial condition y(t=0)=E and y'(t=0)=0. I have formulated the following code:

syms y(t) A B C D E

Dy= diff(y,t);

D2y= diff(y,t,2);

ode = A*D2y + B*sin(C*y) + D*Dy == 0;

cond = y(0)== E;

cond2 = Dy(0)==0;

ySol(t) = simplify(dsolve(ode,conds))

The output says unable to find explicit solution. I'm unsure what to do further to solve it.

[madhan ravi]

Maybe use numerical methods using ode solvers

Answer 1

[Stephan]

Hi,

numeric solution you get by choosing values for A-D and the initial conditions. Then use for example:

syms y(t)
A = 5;
B = 1.5;
C= 3;
D = 25;
ode = A*diff(y,t,2) + B*sin(C*y) + D*diff(y,t) == 0;
[odes, vars] = odeToVectorField(ode);
odefun = matlabFunction(odes,'Vars',{'t','Y'});
y0=[-5 3];
tspan = [0 3];
[t, ySol] = ode45(odefun,tspan,y0);
plot(t,ySol(:,1),t,ySol(:,2))

Note that, since this is a second order ode you need 2 initial conditions for y(t) and Dy(t).

Best regards

Stephan

Answer 2

[Star Strider]

Your function is nonlinear, and most nonlinear ODES do not have analytical solutions.

Try this:

syms y(t) A B C D E Y 
Dy= diff(y,t);
D2y= diff(y,t,2);
ode = A*D2y + B*sin(C*y) + D*Dy == 0;
[VF,Subs] = odeToVectorField(ode)
odefcn = matlabFunction(VF, 'Vars',{t, Y, A, B, C, D, E})

Then provide numerical values for the constants (A, B, C, D, E), and use it as an argument to one of the numeric ODE solvers, for example:

tspan = [0 42];
Y0 = [0, 1];
[T,Y] = ode45(@(t,Y)odefcn(t, Y,A, B, C, D, E), tspan, Y0)

You may need a ‘stiff’ solver, such as ode15s, if the constants have widely-varying magnitudes.