In this segment, we'll talk about how to solve a first order ordinary differential equation exactly
and we're going to take the case where we will be forced to take the next independent solution
for the particular part. So, you will come to know about it when we take the example.
So, here is the example which we are going to be
solving. This is the differential equation given to us with the initial condition at
0 to be 5. Let’s go ahead and see how we're going to go about solving it by using the
so-called classical solution techniques where we find the homogeneous part and the particular part,
so, your full solution becomes the addition of the homogenous part and the particular. Part.
Let's go ahead and concentrate on the homogeneous part first. So, the homogeneous part can be found
by writing the characteristic equations, will be 2 times m to the power 1 because it's the
first derivative plus 3, 0th derivative, m to the power 0 equal to 0. So, 2 m plus 3 is equal to 0,
so m is equal -1.5. So, that becomes the root of the characteristic equation which allows us to say
that, hey the homogeneous part of the solution will be k*e to the power mx,
so you get k*e to the power of 1.5 x. So, that is the homogeneous part of the solution,
in the next slide we're going to concentrate on finding the particular part of the solution.
Let's recall the differential equation which we had it was as follows,
and we just found out that the homogeneous part of the solution is k*e to the power -1.5 x.
So, let's see what the particular part of the solution is. The particular part of the solution
is the form of the right-hand side, or the forcing function, and all its possible
derivatives. Since the form of all the possible derivatives e to the power a*x, for example,
is e to the power a x itself, the only term which we need to then consider is the term, which is in
the forcing function, which is e to the power of -1.5x, so far, the form is concerned. But
there's an issue here because the particular part of solution is same as the homogeneous part
of the solution or it's part of the homogeneous part of solution, we cannot accept this as the
particular part of solution because when we if we substitute this into our differential
equation right here on the left side, we get 0 and that won't allow us to find the value of a.
So, we have to look for the next independent solution, so we got to look at a*x*e to the power
of -1.5x as the particular part of the solution. So, let's go ahead and see if we put that
in our differential equation, so we're going to find the value of a by substituting the particular
part in there. So, we get 2 d/dx of a*x*e to the power -1.5x plus 3a*x*e to the power -1.5x. We're
going to take the derivative of this function here which is a product of two functions of x.
So, using that information, we get a 2a*x*e to the power -1.5x because the derivative of x is one*x*e
to the power -1.5x because the derivative of e to the power -1.5x is -1.5 e to the power -1.5x.
So, we get plus 3 a*x*e to the power -1.5x equal to e to the power -1.5x.
So, let's go and expand this and see what we get.
You're going to find out this term is negative of that term, so that's going to
cancel out That's going to allow us to say that hey,
this is the case. For this to be true for every value of x
2a will be equal to 1 and that will imply a is equal to one half.
So, since a is one half that means that the particular part of solution which was of this form
is now one half same as a*x*e to the power -1.5x, which is the same as 0.5a*x*e to the power -1.5x.
So, what we found out was the homogeneous is k*e to the power -1.5, and we found the particular
part to be 0.5x*e to the power -1.5x. So, the complete solution will be k*e to the power minus
1.5x plus 0.5x*e to the power -1.5x. And so, the question arises that, hey, how do I find the value
of k? You find the value of k by applying the initial condition which is y0 is equal to 5.
So, 5 will be equal to the value of the y at x equal to 0,
and that gives me 5 is equal to k this part is 0 that's k, so k is equal to 5.
K is equal to 5, the complete solution is 5e to the power -1.5x plus 0.5x*e to the power -1.5x.
And that is the solution which you are looking for and that's also the end of this segment.