How to build a graph of the solution of a second-order differential equation in MATLAB

I need to build a graph for this second order differential equation in MatLab:

φ(t) + (d²φ(t)/dt²) - ((d²φ(t)/dt²) * m * sin(pi) * a) = m * g * sin(pi) * a.

Where a = v * t, v = 1 and m = 0.1.

0 < t < 1 / v.

I tried this:

if true
v = 1;
m = 0.1;
g = 9.81;
syms t phi(t)
eqn = phi(t) + diff(phi, t, 2) - diff(phi, t, 2) * m * sin(pi) * (v * t) == m * g * sin(pi) * (v * t);
phiSolution(t) = dsolve(eqn);
disp(phiSolution(t));
t = linspace(0, 1 / v, 100);
phiValues = double(subs(phiSolution, t));
plot(t, phiValues);
xlabel('t');
ylabel('φ(t)');
title('Graph of φ(t)');
end

Answer 1

[Torsten]

Since sin(pi) = 0, your differential equation reduces to

y'' + y = 0

with general solution

a*sin(t) + b*cos(t)

Now incorporate your initial/boundary conditions and you are done.

If you meant "phi" instead of "pi", use ode45.

[Mohamad]

Thank you, and how can I implement this please in Matlab?

[Torsten]

v = 1;
m = 0.1;
g = 9.81;
fun = @(t,y)[y(2);(m*g*sin(y(1))*v*t-y(1))/(1-m*sin(y(1))*v*t)];
tspan = [0 10];
y0 = [0 1];
[T,Y] = ode45(fun,tspan,y0);
plot(T,Y(:,1))

[Mohamad]

Thanks

Answer 2

[Steven Lord]

Solve the system of ODEs numerically (try using ode45 first, and switch to a stiff solver if it takes too long) rather than symbolically.

Symbolic Math Toolbox has functions to generate the function for use with the numeric ODE solvers from a symbolic expression.

[Mohamad]

Thanks, I tried ode45 also, but it prints errors too. Can you please provide a code?

[Steven Lord]

Can you show us the code you wrote that tries to solve the problem with ode45 and the full and exact text of the error message you received (all the red text displayed in the Command Window when you ran your code) when you ran that code?