Question

I'm trying to solve this problem by using matlab. But I don't know reason why I can't get the solutions. I wanna get a plot of this differential equation. Please find a way how to solve this problem. May there're errors in the code. Please check it.   

second-oder-ode2.m x 曱function, second-oder-ode2 t=0:0.001 :30; initial-× = 0; in i t i al-dxdt 0; lt,影=ode45( @rhs, t. [initial.x initial-dxdt ] ); plot( (:, 1) ) ; xlabel( 't); ylabel(): function dxdt=rhs( t, x) dxdt 1 x(2) dxdt.2 (-50 x(2) 61.25+((1-cos(4 pi 10 t))/2) (47380 x(1) 3-7428-x(1)*2+366*x(1)-7): end end 오후 10:1 닛 れな. -56 166 Warning: Failure at t-2.606178c-01. Unable to meet integration tolerances without reducing the step size below the smallest value allowed (8.881784c-16) at time t

Answer 1

The reason why the system is not able to evaluate the integral is that the system dynamics are unstable and the response is growing exponentially, evident from the simulation that I have carried out. The response is plotted below.

-50 -100 X -150 200 -250 0.005 0.02 0.01 0.015 0.025 0.03 0.035 t

From the above response, it is observed that the system is unstable.

I have simulated the same system with a slight modification to the dynamics by replacing the positive sign with a negative sign before the second term of dxdt_2 (i.e. - 61.25). The code is given below with the response.

clc;
close all;
clear all;

t = 0:0.001:30;

initial_x = 0;
initial_dxdt = 0;

[t,x] = ode45(@rhs,t,[initial_x initial_dxdt]);

plot(t,x(:,1));
xlabel('t');ylabel('x');

function dxdt = rhs(t,x)

dxdt_1 = x(2);
dxdt_2 = -50*x(2) - 61.25*((1-cos(4*pi*10*t))/2*(47380*x(1)^3-7428*x(1)^2+366*x(1)-7));
dxdt = [dxdt_1 ; dxdt_2];
end

The stable response is plotted below.

0.1 0.09 0.08 0.07 0.06 x 0.05 0.04 0.03 0.02 0.01 0 10 15 t 30 25 20 LO