2nd order systems of differential equation

• d2y1/dt2+5*dy1/dt+y2^2-y1=0;
• d2y2/dt2+5*dy2/dt+y3^2-y2=0;
• d2y3/dt2+5*dy3/dt-y3=0;

I tried my level best, however i could not able to solve it. my errors are undefined function of x and double. running errors.

    % code
function dydt=hi(t,y) %#ok<INUSL>
dydt=zeros(6,1);
n=3;
dydt(1:n) = y(n+1:2*n);
for i=n+1
    dy(i)=x(i);
    if i<2*n
        dydt(i)=-5*x(i)-y(i+1)^2+y(i);
    else
        dydt(i)=-5*x(i)+y(i);
    end
end  
    % code

[Azzi Abdelmalek]

What x is doing in your code?

Answer 1

[Azzi Abdelmalek]

By adding 3 variables y4,y5 and y6 your system becomes a first order differential equations system.

%y4=dy1/dt, thus d2y1/dt2=dy4/dt
%y5=dy2/dt, thus d2y2/dt2=dy5/dt
%y6=dy3/dt, thus d2y3/dt2=dy6/dt

%The system of equation becomes

%dy1/dt=y11; 
%dy2/dt=y22; 
%dy3/dt=y33;
%dy4/dt+5*y4+y2^2-y1=0;
%dy5/dt+5*y5+y3^2-y2=0;
%dy6/dt+5*y6-y3=0;

% which can be programmed

function dy=myequ(t,y)
dy(1)=y(4)
dy(2)=y(5)
dy(3)=y(6)
dy(4)=-5*y4-y2^2+y1
dy(5)=-5*y5-y3^2+y2
dy(6)=-5*y6+y3

% you can call myequ

tspan=[0 10];
y0=[1 0 1 0 1 1]; % Initial conditions
[t,y]=ode45(@myequ,tspan,y0)

[lenin]

Hi Azzi, thanks for your answer. i did take 'x' wrongly to use derivative of dydt. thansk for correcting me and that code is running. however, i am going to use below one due to i have if more than 3 equations.Thanks for your help.

    % n=3;
% dy(1:n)=y(n+1:2*n);
% for i=n+1:2*n
%     if i<2*n
%        dy(i)=-5*y(i)-y(i-2)^2-y(i-3);
%     else
%         dy(i)=-5*y(i)-y(i-3);
%     end
% end

Answer 2

[Susmita panda]

I have a differential equation of 2nd order: D2q(t)+ 2 zeta*w*Dq(t)+w^2*q(t)=2p(1+k)/(mL(1+8k/pi+2k^2))(sin(pi*v*t/L)+k) with initial conditions q(0) and Dq(t) = 0

I am getting a very lengthy solution. Is there any way to either simplify it or to confirm the differential equation from solution? Kindly suggest better way to solve such differential equation.

 %code
syms q(t) zeta w P k m L v; 
F0=2p(1+k)/(mL(1+8k/pi+2k^2))*(sin(pi*v*t/L)+k) ;
Dq=diff(q);
ode=diff(q,t,2)+2*zeta*Dq+w^2*q(t)==F0;
cond1=q(0)==0;
cond2=Dq(0)==0;
conds=[cond1 cond2];
ySol(t)=dsolve(ode,conds);
ySol=simplify(ySol)