# Finding the closed form solution to a system of second order differential equations

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Aleem Andrew on 19 Mar 2020
Commented: Star Strider on 22 Mar 2020
I am trying to find the closed form solution to a system of differential equations but I am unsure how to access the closed form solutions. How can I modify the code to find the closed form solutions of y and z? Any help would be appreciated.
syms y(t) z(t)
Dy = diff(y,t); %angle
D2y = diff(y,t,2);
Dz = diff(z,t); D2z = diff(z,t,2);
eqns = [-4.905*sin(y)-24.525*sin(y) == 0.046875*D2y + (0.125*sin(y)*(Dy*Dy*cos(y)+D2y*sin(y))) + sin(y)*(D2y*0.625*sin(y))-D2y*cos(y), D2z == Dy*Dy*0.25*sin(y)-D2y*0.25*cos(y)];
fun = matlabFunction(odeToVectorField(eqns),'Vars',{'t','Y','Z'});
tspan = [0 10]; % Time interval for integration
y0 = [pi/3 0 0 0]; % initial conditions
%[t,y] = ode45(fun,tspan,y0);
sol = ode45(fun,[0 20],y0)
fplot(@(x)deval(sol,x,1), [0, 20])

Star Strider on 19 Mar 2020
A closed-form solution is likely not possible because both differential equations are nonlinear, specifically:
(0.125*sin(y)*(Dy*Dy*cos(y)+D2y*sin(y))
and:
Dy*Dy*0.25*sin(y)-D2y*0.25*cos(y)
Most nonlinear differential equations do not have analytic solutions. These are two of them.
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Star Strider on 22 Mar 2020
As always, my pleasure!