In the tutorial How to solve an ordinary differential equation (ODE) in Scilab we can see how a first order ordinary differential equation is solved (numerically) in Scilab. In this tutorial we are going to solve a **second order ordinary differential equation** using the embedded Scilab function `ode()`

.

As example we are going to use a nonlinear second order ordinary differential equation:

\[\frac{d^2 y}{dt^2} = \frac{1}{t+1} + sint(t)\sqrt{t}\]with the initial conditions:

\[ \begin{equation*} \begin{split}y(0) &= 0\\

\frac{dy}{dt}(0) &= -2

\end{split} \end{equation*} \]

The first step is to write the second order differential equation as a system of two first order differential equations:

\[ \begin{equation*} \begin{split}\frac{dy}{dt} &= z\\

\frac{dz}{dt} &= \frac{1}{t+1} + sint(t)\sqrt{t}

\end{split} \end{equation*} \]

This way we have set our differential equation in the form:

\[\frac{dx}{dt}=f(t,x)\]where both *dx* and *x* are matrices of size `2 x 1`

.

x &= \left [ z; \frac{1}{t+1} + sint(t)\sqrt{t} \right ]\\

dx &= \left [ \frac{dy}{dt}; \frac{dz}{dt} \right ]

\end{split} \end{equation*} \]

To solve this differential equation in Scilab, first we need to define our differential equation as a separate function. Scilab allows to define a custom function is an `*.sce`

file, together with other instructions. For this example, all of the Scilab instruction will need to be included in the same `*.sce`

file.

**Step 1**. Define the differential equation as a custom function

function dx = f(t, x) dx(1) = x(2); dx(2) = 1/(t+1) + sin(t)*sqrt(t); endfunction

**Step 2**. Define the integration time *t*, initial time *t _{0}* and initial values

t = 0:0.01:5*%pi; t0 = min(t); y0 = [0; -2];

**Step 3**. Call the `ode()`

function with the above parameters

y = ode(y0, t0, t, f);

**Step 4**. Plot the result. Notice that the numerical solution `y`

contains the second and first integration results, *y* and *y’*.

plot(t,y(1,:),'LineWidth',2) plot(t,y(2,:),'r','LineWidth',2) xgrid(); xlabel('$t \quad [s]$','FontSize',3) ylabel('$f(t,x)$','FontSize',3) title(['Integration of ' '$\frac{d^2 x}{dt^2} = \frac{1}{t+1} + sint(t)\sqrt{t}$'],'FontSize',3) legend(['$\Large{x}$' '$\Large{dx/dt}$'],2)

By running all of the above Scilab instruction in an script file (`*.sce`

), we get the following graphical window:

To check that the solution of our integration is correct, we are going the model the equation in Xcos and run the simulation for `15.71`

seconds (5π).

The Xcos block diagram model of the second order ordinary differential equation is integrated using the **Runge-Kutta 4(5)** numerical solver.

After running the simulation, **Xcos** will output the following graphical window (the grid has been added afterwards):

As you can see, both methods give the same results. This is a confirmation that the system of first order ODE were derived correctly and the equations were correctly integrated.

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