Let me introduce
the following concept.
Consider the differential equation for a gain,
which is linear nth
order homogeneous differential equation,
say a_sub_n of x times nth derivative of y plus a_sub_(n-1) of x times
(n-1) derivative of y plus... a sub 1 of
x times y prime plus a sub 0 of x times y that is equal to 0.
Linear nth order variable coefficient homogeneous differential equation,
where I assume that all
the coefficients ai of x,
they are continuous and
sum interval I and the leading coefficients a_sub_n of x,
is never 0 on the interval I.
For this differential equation (4),
we call any set of
n linearly independent solutions of that differential equation on an interval I,
we call it a fundamental set of
solutions of differential equation (4) on the interval I.
You must note that the order of this differential equation which is n,
it must coincide the number of linearly independent solutions.
We should have exactly n linearly independent solutions of that differential equation.
Then we call the set y_sub_i,
where i is moving from 1 to the n,
to be a fundamental set of solutions of differential equation (4) on this interval.
Any such a differential equation always has
a fundamental set of solutions as to following theorem shows.
Existence of a fundamental set of solutions.
Any linear homogeneous differential equation (4), L(y) = 0.
I hope you will remind what is L, L(y),
it's the nth order linear differential equation,
always has a fundamental set of solutions on I.
It always have a fundamental solution.
In proving this theorem,
again I'm assuming that n = 2 to make the thing simple.
And by Theorem 4.2, that's Theorem 4.2,
it's a unique distance of solution for initial value problem.
So by that theorem,
choose arbitrary point x0 in the interval
I and consider the following two initial value problems.
Differential equations are the same thing.
The second order, linear homogeneous differential equation.
First initial value problem requires the initial condition to
be y(x0) = 1 and y prime (x0) = 0.
Second initial value problem requires y(x0) = 0,
y prime (x0) = 1.
All this first and second initial value problems.
By Theorem 4.2, they should have a unique solutions,
say y_sub_1 and the y_sub_2 on the interval I, right?
That is guaranteed by the Theorem 4.2, right?
For these two solutions of y1 and y2,
y1 is a solution to this initial value problem first,
y2 is solution to second initial value problem.
Compute the Wronskian at point (x0).
That's the determinant of these 2 by 2 matrices.
How much is this one?
That is 1.
How much is this one?
Because this is a solution to the second problem,
that is 0. What is the y1 prime of (x0)?
That is 0 from this initial condition. What is the y2 prime of (x0)?
That must be equal to one by this initial condition.
So, our 2 by 2 matrix is 1, 0, 0, 1.
Its determinant 1, which is not equal to 0, right?
So, by the Theorem 4.3 that we've just proved,
these two solutions y_sub_1 and y_sub_2 are linearly independent on I,
and the solutions of differential equation.
Because this is a second order and we now have
two linearly independent solutions that we call a fundamental solution of the equation.
So we are done.
At least for little n is equal 2.
We confirm that there must to be
a fundamental set of solutions.
For example, in example 1 we handled before,
we have seen that y_sub_1 = x and y_sub_2 = x times the log of x.
Both of them are solutions of this variable coefficients, linear,
second order, homogeneous differential equation,
on the interval from 0 to infinity.
Can you remind it.
That's a computed the Wronskian.
Wronskian of W(y1, y2), what is it?
This is y1 which is x and y2,
that is X times the log of x,
and its derivative is 1,
its derivative of X is log (x + 1).
Then how much is the determinant?
X log of x plus x,
minus X log of x that is X, right?.
This is never 0 on the interval from 0 to infinity,
right? What does that mean?
By the Wronskian test,
the two solution x,
and x times the log x,
is a fundamental set of solutions to
given homogeneous second order differential equation on the interval from 0 to infinity.
We can confirm it.
