solve a system of nonlinear differential equations whit initial conditions

i need to solve the system diff(xl,t,2)+(beta*(1+xl^2)*diff(xl,t))+xl +(delta/2*xl) == alpha*(diff(xr,t)+diff(xl,t)) diff(xr,t,2)+(beta*(1+xr^2)*diff(xr,t))+xr -(delta/2*xr) == alpha*(diff(xr,t)+diff(xl,t)) with the conditions xl(0) == 1 xr(0) == 1 xl'(0) == 0 xr'(0) and the parameters: alpha=0.5;beta=0.3; delta=0.

i try with:

% inizialize patameters
alpha=0.5
beta=0.3
delta=0;

syms xl(t) xr(t) ;
eqn1 = diff(xl,t,2)+(beta*(1+xl^2)*diff(xl,t))+xl +(delta/2*xl) == alpha*(diff(xr,t)+diff(xl,t));
eqn2 = diff(xr,t,2)+(beta*(1+xr^2)*diff(xr,t))+xr -(delta/2*xr) == alpha*(diff(xr,t)+diff(xl,t));
eqns =[eqn1, eqn2];

dl= diff(xl,t);
dr= diff(xr,t);
cond1 = xl(0) == 1;
cond2 = xr(0) == 1;
cond3 = dl(0) == 0;
cond4 = dr(0) == 0;
cond = [cond1, cond2, cond3, cond4];
[Sol_xl(t),Sol_xr(t)]= dsolve(eqns,cond);

but goves me the errors:

Warning: Unable to find symbolic solution.

> In dsolve (line 209)

In (line 20)

Error using sym/subsindex (line 857)

Invalid indexing or function definition. Indexing must follow MATLAB indexing. Function arguments must be symbolic variables, and function body must be sym expression.

Error in (line 20)

[Sol_xl(t),Sol_xr(t)]= dsolve(eqns,cond);

Answer 1

[Ishan]

If dsolve cannot find an explicit solution of a differential equation analytically, then it returns an empty symbolic array. You can solve the differential equation by using MATLAB numerical solver, such as ode45. For more information, see Solve a Second-Order Differential Equation Numerically.

Likely the only way to solve it is to use odeToVectorField and matlabFunction functions to create an anonymous function to use with the numeric ODE solvers, such as ode45 or ode15s:

You may start with this for your case : -

alpha=0.5;
beta=0.3;
delta=0;
syms y(t) x(t)
eqn1 = diff(x,t,2)+(beta*(1+x^2)*diff(x,t))+x +(delta/2*x) == alpha*(diff(y,t)+diff(x,t));
eqn2 = diff(y,t,2)+(beta*(1+y^2)*diff(y,t))+y -(delta/2*y) == alpha*(diff(y,t)+diff(x,t));
eqns =[eqn1, eqn2];
[V,S] = odeToVectorField(eqns)

V = 

S = 

M = matlabFunction(V,'vars', {'t','Y','X'})

M = function_handle with value:
    @(t,Y,X)[Y(2);-Y(1)+Y(2)./2.0+Y(4)./2.0-Y(2).*(Y(1).^2.*(3.0./1.0e+1)+3.0./1.0e+1);Y(4);Y(2)./2.0-Y(3)+Y(4)./2.0-Y(4).*(Y(3).^2.*(3.0./1.0e+1)+3.0./1.0e+1)]

%[t,y] = ode45(odefun,tspan,y0) %configure the ode solver as per your required given initial values.