In this segment, we'll talk about how we can take coupled ordinary differential equations
and reduce them to first order ordinary differential equations. That forms the
background of this particular segment because if we're going to solve any ordinary differential
equation by using numerical methods, we can only solve first order ordinary differential equations.
And if we are given couple ordinary differential equations and they have a higher order in them,
then we need to be able to reduce them to first order ordinary differential equations.
So let's go and do this to an example. So let's suppose this is the first ordinary differential
equation given to us, this is the second ordinary differential equation given to us.
They are coupled in nature, or what we may call a simultaneous ordinary differential equations.
As you can see, the dependent variable x1 and x2 exists in both of the differential equations so
x1 and x2 are the so x1 and x2 are the dependent variables and T is the independent variable. And
these are the initial conditions which have been given for these two coupled ordinary
differential equations. So what we want to do is we want to be able to take the above couple
ODEs and reduce them to a set of simultaneous first order ODEs so that numerical methods such as
Euler's method or Runge-Kutta second order method when you go to fourth order method can be applied
to be able to solve these differential equations. So what we're going to do is we're going to take
one differential equation at a time and see how we can reduce it, and that's the way to follow it.
So this is the first differential equation which is given to us right here and we're already seeing
that hey this one is second order. So what we're going to do is we're going to take dx1 by dt,
and we can introduce a new variable called let's suppose v1. And what that's going to do is that
when I take the first derivative of dx1 over dt, I will get the first derivative of v1, hence
implying that hey somehow I'm able to convert this to a first order ODE. Let me also introduce that
dx2 by dt will be equal to v2, and that makes it that the second derivative of x2 with respect to
time, or t, is dv2 by dt. This will be useful in our second ordinary differential equation,
but we have to make the substitution of this one right here in this differential equation as well.
So as you can see that if I rewrite this differential equation now, I get 5,
I get dv1 by dt because that's what it is plus 3 and, well for this one I'll substitute v1,
and for this I'll substitute v2. And plus 7x1 minus x2, nothing changes there, and 29cos(31t).
So having said that, what we're going to do is we're going to rewrite this in the form of,
if you remember all the differential question results so far of the form dy by dx is equal to
f(x,y). So that's what we're going to do. We're going to take dv1 by dt right here and that we
take this to the right side. Okay, take this to the right side and then we have 29cos(31t),
and we're going to divide it by 5 here. So we are writing rewriting this differential equation
in the form so that we can solve by using our numerical methods. So from here I get this and
this and
so that reduces our second order ordinary differential equation to the first one, but keep
in mind that we are not simply magically trying to convert a second order differential equation to
a first order. We will have to introduce another differential equation which will be of this form
so that it this second order differential equation reduces to first order ordering
differential equations. We'll write that later, but the first thing which we need to understand
is that hey what is the corresponding initial condition to this one. And this is what students
get into some issues they don't understand. What is the corresponding initial condition to this
one? But you can understand that if you look at from a pragmatic point of view is that hey if I
know the current value of v1 and I know the slope of v1, I can find out what the value of v1 will
be at a point ahead. And that's why the first that the initial condition has to be on v1. So
v1 of 0 is same as dx1 by dt of 0, and that was given to us as 41. So that takes care of this
differential equation and now what we're going to do is we're going to take care of the next one.
So this is our second order ordinary differential equation and this is the second of those second
order differential equations. So now here again let me rewrite this again. We had dx1
by dt equal to v1, so I'm not introducing anything new but I'm rewriting it here so
that we can do the proper substitutions and see where the substitutions are coming from.
So we had to introduced two new dependent variables, v1 and v2, in the previous slide
and we are just repeating what we had done there. So having said that, what we're going
to do is this one will become 10 and this is the second derivative of x1 with respect to T,
this is second derivative with respect to x2, so we'll get dv2 by dt here. Then 13 because dx2
by dt is v2, so I'll get v2. This one is v1 and that's how we're going to rewrite our equation.
So once we have done that, we got to rewrite it in the form which can be which can allow
us to use numerical methods so we will just say write down the equation for dv2 by dt
so I'll have to take this to the right side and this to the right side and this is the
right right side anyway divided by 10, and that gives me a simplified this version right here.
So this gives us the now you can see that hey this differential equation to a second order in
nature has been reduced to a first order ordinary differential equation, but at the same time you
got to understand that we have to couple it with another differential equation, which will be this
one, that dx2 by dt is equal to v2. People might think that hey this is introduced variable but
this itself is an ordinary differential equation and that's one what's going to help us to reduce
the second order ordinary differential equation to a set of first order ordinary differential
equations, we'll see that in the next slide but this is what you have to think about it here.
Now the initial condition will be on v2 here. So what is v2 of 0? v2 of 0 is whatever dx2 by dt of
0 is and we so we have been given that number to be 47. So what we got to think about is that we
have this differential equation right here and, corresponding to the initial condition here,
we have this differential equation right here. The corresponding initial condition will be on x2 of
course for that one, so that gives us 2 in the previous slide. Also we had 2 so we have reduced
our 2 second order simultaneous second order first second order differential equations to
4 first order ordinary differential equations. So let's go and see that how that works out.
So if we look at the first order ordinary differential equations which we derived,
so this was the first one which you derived right here and now this is a function of the 4 deep end
variables, x1, x2, or the original dependent variables, v1 and v2 are the ones which we
introduce, and T which is the independent variable right here. And the corresponding
initial condition of course will be on the dependent variable of that first order ODE,
which is v1 in this case. Same thing, this was the second ODE which we converted into
first order ordinary differential equation, so it's be written in that particular form so that
we can write it as a function of the dependent variables again and the independent variable T,
corresponding initial condition will be on the dependent variable v2, which is given as 47.
Now we have reduced our second order ordinary differential equation to the first order,
but we have to have 4 first order ordinary differential equations. So this one, although
this is looks like hey this is the variable which we introduced in order to set up the first order
order interfacial equation, this itself is a differential equation because it is a function of
the four dependent variables and one independent variable right there and the corresponding initial
condition will be of x1, so that will be 37. Same thing right here, we introduce this dependent
variable v2 here but this itself is an ordinary differential equation and it's a function of
the four different variables and the independent variable T and the corresponding initial condition
will be on the different variable x2, which is 43 here. So you basically have 1, 2, 3, and 4
first order ordinary differential equations which you will need to solve simultaneously
by any numerical method of whatever of your choice or whatever is being asked of you to do. It can be
the Euler's method, it can be Runge-Kutta second order method, Runge-Kutta fourth order method,
but this is the way how you go about rewriting your original ordinary differential equations
so that they can be solved by using numerical methods. And that's the end of this segment.