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

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?

Answer 1

[Star Strider]

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.

.

[Shreyas Sangamesh]

Hello,

thank you helping me to solve this problem,

But, we need to use ode45 instead of ode89 right?

Because when we try to solve with ode89 probably it won't plot, I tried with ode45 it works!!

Let me know your views on this!!

[Star Strider]

I just wanted to see what the higher order integrator could do. Use whatever integrator you want.

[Shreyas Sangamesh]

Okay perfect,

Thank you for assistance.

[Star Strider]

As always, my pleasure!