# Plotting 2nd order ode with time dependent parameters

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Jake-Anthony Vickers on 12 Oct 2021
I need to use the ODE45 function to plot the following ODE (shown below) with the initial conditions𝜃(0)=𝜋/2 rad and 𝜃(𝑡)=0.1 rad/s and 0 ≤ 𝑡 ≤ 10 seconds. One plot for𝜃(𝑡) when 𝛺=𝜋 and another when 𝛺=4𝜋.
theta'' + theta' + 10*cos(2pi*theta) = sin(omega*t)
Im having difficulty understanding how to create a function that can solve this using ode45. I was able to solve earlier ones when it was a simple linear 2nd ode- Homogeneous. Below is how I solved the other 2nd order.
function = second_order
%% y'' + 6y' + 9y = 0 initial function
yinitial = [4, 0];
t = 0:0.001:3;
[T, STATES] = ode45(@ode2_hw8, t, yinitial);
Y = STATES(:,1);
Ydot = STATES(:,2);
figure
subplot(1,2,1)
plot(T,Y)
subplot(1,2,2)
plot(T, Ydot)
end
function [out2] = ode2_hw8 (t, state)
b = 6;
k = 9;
y = state(1);
ydot = state(2);
yddot = -b*ydot - k*y;
out2 = [ydot; yddot];
end

Ashutosh Singh Baghel on 20 Oct 2021
Hi Jake,
I believe you wish to perform a second order differential ODE with time dependent parameter. Please refer to the following example -
% Second Order ODE with Time-Dependent Terms
% \$\$ y''(t) +y'(t) +10*cos(2*pi*y) = sin(w*t);
% w = omega = pi, 4*pi, etc
% t = (0,10)
% The initial condition is \$y_0 = [pi*0.5 , 0.1]\$.
omega = pi;
gt = linspace(0, 10, 50);
g = sin(pi*gt);
tspan = [0 10];
y_0 = [pi*0.5 0.1];
[t,y] = ode45(@(t, y) myode1(t, y, gt, g), tspan, y_0);
plot(t, y(:,1), '-r ', t, y(:,2), '--g');
legend('y(1)', 'y(2)');
xlabel('t');
ylabel('y Solution'); function dydt = myode1(t, y, gt, g)
g = interp1(gt, g, t); % Interpolate the data set (gt, g) at time t
dydt = [y(2); -(y(2)+10*cos(2*pi*y(1))) + g]; % Evaluate ODE at time
end
Please refer to the following MATLAB Documentation page on the 'ode45' function and relevant information on the 'Differential Equation'.

R2020b

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