How do I solve this differential equation 2nd order numerically within matlab?

Hello everyone i am completly new to Matlab and I have a problem with an exercise in my practice book. I have this differential equation:

y(0) = 1 dy(0)=0

and I would like to solve it numerically. I could also solve it simbolically but i already know how to do that and i want to practice.

So first i have to change it in first order differential equation:

But how do i continue? I´m trying for hours at this point😅

Answer 1

[Sulaymon Eshkabilov]

There are a couple of points.

(1) As you showed, it is an IVP. But you have one Initial Condition y(0), what about dy(0) =?

(2) What is in sigma(t)?

You can use ode45 - see DOC that shows how to solve similar exercises.

Step 1. Define a fcn file or anonymous fcn handle, e.g.:

F = @(t,y)([y(2); 2*sigma(t) - 3*y(2) - y(1)]);

Step 2. Set up initial conditions:

IC0 = [1; ???];

Step 3. Set up time span to simulate your exercise

t = [0, 10];

Step 4. Solve

[Time, YSol] =ode45(F, time, IC0);

Step 5. Plot the results

plot(Time, YSol(:, 1)), ...

[Karl-JR]

oh sorry i forgot dy(0) i edited it

[Sulaymon Eshkabilov]

what about sigma(t)?

[Karl-JR]

It´s not given

[Karl-JR]

I got sigma(t) through solving it first simbolically. and now have this so far (func4 is the equation broken into 2 first orders :

yo = [1 0];
tspan = 0:.1:10;
[t,y] = ode45 ('func4',tspan,yo);
plot(t,y(:,1))
hold on
plot (t,y(:,2))
hold off

did i make a mistake here? It says the "Index exceeds number of array elements."

Answer 2

[Sulaymon Eshkabilov]

Here is a corrected complete code:

F = @(t,y)([y(2); 2*sin(t) - 3*y(2) - y(1)]);  % Note signa(t) = sin(t)
IC0 = [1; 0];                   % Initial conditions
tspan = [0, 10];                % Time span  
[t,ysol] = ode45(F,tspan,IC0);
plot(t,ysol(:,1),'b-', 'DisplayName', 'y(t)')
hold on
plot (t,ysol(:,2), 'r-', 'DisplayName', 'dy(t)')
legend show
xlabel('t, [s]')
ylabel('y(t), dy(t)')
grid on
hold off