How to solve 2nd ODE equation in numerical and analytical method at same plot graph?

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Hello,
I am was looking for help to solve this equation
d^x/dt^2 + dx/dt + x =1
in Matlab, can I get help to solve this equation of plot the numerical solutions and analytical solutions in the same graph and compare them.
with ode45
can anyone help me with this code?

Accepted Answer

Star Strider
Star Strider on 28 Sep 2022
Interesting problem!
I was curious enough to do the entire simulation to see what the results are —
syms x(t) x0 Dx0 T Y
Dx = diff(x);
D2x = diff(Dx);
DEqn = D2x + Dx + x == 1;
xs = dsolve(DEqn, x(0)==x0, Dx(0)==Dx0)
xs = 
xs = subs(xs, {x0,Dx0},{1,1}) % Numerical Initial Conditions
xs = 
[VF,Sbs] = odeToVectorField(DEqn)
VF = 
Sbs = 
DEfcn = matlabFunction(VF, 'Vars',{T,Y})
DEfcn = function_handle with value:
@(T,Y)[Y(2);-Y(1)-Y(2)+1.0]
tspan = [0 10];
[t,x] = ode89(DEfcn, tspan, [1 1]);
figure
fplot(xs, tspan, 'DisplayName','Analytic')
hold on
plot(t, x(:,1), 'DisplayName','Numeric')
hold off
grid
xlabel('Time')
ylabel('x_1(t)')
legend('Location','best')
They match almost exactly.
.
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