2nd order differential eqation.

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##### Question

Dear sir/madam

i am trying to learn the 2nd ODE, i am not able to understand the derivation of value of dx(1)=x(2); in first 4 line of function,

even i have see this x(2) component in many other available example for different, case but this part is same in all equation,

so i request your to explain me the role of dx(1);

i am trying to learn the 2nd ODE, i am not able to understand the derivation of value of dx(1)=x(2); in first 4 line of function,

even i have see this x(2) component in many other available example for different, case but this part is same in all equation,

so i request your to explain me the role of dx(1);

Scilab ODE-Applications 03-04 min 40-50 sec

__Answers:__

x(1), x(2) refer to the variables. The variables are represented as a vector x (or y or any variable of your choice) . For example, in motion of simple pendulum, the equations are

theta- Angle made by the pendulum at time t.

d(theta)/dt=theta'; (first derivative)

d(theta')/dt=theta'';(second derivative)

It is also given,

theta''=(9.8/0.5)*sin(theta);

Let us rewrite these equations with a vector x. We are rewriting these equations to convert the system of second order ODEs into a system of first order ODEs.

Let x(1)=theta;

Let x(2)=theta';

x(2)=theta'=d(theta)/dt=dx(1);

Let x(3)=theta'';

x(3)=theta''=d(theta')/dt=dx(2);

dx(2)=x(3);

Rewriting these equations,

dx(1)=x(2);

dx(2)=(9.8/0.5)*sin(x(1));

Now we have a system of first order ODEs. I hope this solves your doubt.

theta- Angle made by the pendulum at time t.

d(theta)/dt=theta'; (first derivative)

d(theta')/dt=theta'';(second derivative)

It is also given,

theta''=(9.8/0.5)*sin(theta);

Let us rewrite these equations with a vector x. We are rewriting these equations to convert the system of second order ODEs into a system of first order ODEs.

Let x(1)=theta;

Let x(2)=theta';

x(2)=theta'=d(theta)/dt=dx(1);

Let x(3)=theta'';

x(3)=theta''=d(theta')/dt=dx(2);

dx(2)=x(3);

Rewriting these equations,

dx(1)=x(2);

dx(2)=(9.8/0.5)*sin(x(1));

Now we have a system of first order ODEs. I hope this solves your doubt.

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