# Plotting a function defined by a system of ODEs

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Cris19 on 7 Jun 2018
Commented: Cris19 on 7 Jun 2018
I am trying to solve with MATLAB the first order ODEs system,
$$\left\{ \begin{array}{l} x_{1}^{\prime }=-\frac{1}{t+1}x_{1}+x_{2} \\ x_{2}^{\prime }=-(1+e^{-2t})x_{1}-\frac{1}{t+1}x_{2}+\frac{e^{-3t}}{t+1}x_{3} \\ x_{3}^{\prime }=-\frac{1}{t+1}x_{3}+x_{4} \\ x_{4}^{\prime }=e^{-3t}\left( t+1\right) x_{1}-\left( 1+e^{-2t}\right) x_{3}-% \frac{1}{t+1}x_{4}-\frac{1}{t+1}x_{3}^{2}% \end{array}% \right.$$
I've defined the function:
function dzdt=odefun(t,z)
dzdt=zeros(4,1);
dzdt(1)=-(1/(t+1))*z(1)+z(2);
dzdt(2)=-(1+exp(-2*t))*z(1)-(1/(t+1))*z(2)+(exp(-3*t))/(t+1)*z(3);
dzdt(3)=z(4)-(1/(t+1))*z(3);
dzdt(4)=(exp(-3*t))*(t+1)*z(1)-(1+exp(-2*t))*z(3)-(1/(t+1))*z(4)-(1/(t+1))*z(3)^2;
end
The time interval is [0,100] and the initial conditions are z0 = [0.01 0.01 0.01 0.01].
With the ode45 solver, I've used the commands:
>> tspan = [0 100];
>> z0 = [0.01 0.01 0.01 0.01];
>> [t,z] = ode45(@(t,z) odefun(t,z), tspan, z0);
>> plot(t,z(:,1),'r')
and I've obtained easily the graph of z(1)=x_1.
But I want to plot the function f(t)=(t+1)*x_1(t), t\in [0,100], where x_1=z(1) is the first unknown of the system. How could I do this ?

Torsten on 7 Jun 2018
tspan = [0 100];
z0 = [0.01 0.01 0.01 0.01];
[t,z] = ode45(@(t,z) odefun(t,z), tspan, z0);
z1 = (t+1).*z(:,1);
plot(t,z1)
Best wishes
Torsten.
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Cris19 on 7 Jun 2018
Thank you, Torsten! Best wishes, Cris