Analytical Proof of the Solution to Second Order Linear Homogeneous Differential Equation ()
1. Introduction
Generally, equations are viewed as mathematical statements that relate one or more physical quantities usually referred to as variables. These variables are so called because they undergo changes. Equations that describe the relationship between variables are known as functions. An example of functions is given by the equation below:
(1)
In Equation (1), x and y are variables. Functions describing the rates of change of variables are called derivatives. Equation (2) below is a function which is also a derivative
(2)
In Equation (2),
is the derivative of y with respect to x. Differential
equations are those which relate one on more functions and their derivatives. As noted earlier, the functions represent physical quantities that are subject to change while the derivatives represent their rates of change.
Differential equations occur very commonly in different phenomena, hence their role in many disciplines including such fields as engineering, physical and chemical sciences, economics and biology [1]. An example of differential equations is given by the equation below.
(3)
Equations of the kind above first came into existence through the efforts of Isaac Newton and Gottfried Leibniz who independently pioneered calculus around the 17th century. Others who made significant contributions, to the development of differential equations in the 18th century included Jacob Bernoulli, Jean Le Rond D’Alembert, Leonard Euler, Daniel Bernoulli, Joseph-Louis Legrange, and Laplace. From the 19th century through the 20th and up to the 21st century, outstanding achievements were made by quite a lot of great mathematicians prominent among them were Fourier Maxwell and Pioneare [2].
Reflecting retrospectively on the development of differential equations actually brings to mind the temptation to think that the field of differential equations has been stretched to its elastic limit. In other words, it seems quite likely that the field of differential equations has been exhausted of new topics to explore, leaving nothing to be investigated any further. This probably accounts for why research efforts are grossly directed toward their applications.
Grappling with the foregoing conception, curiosity drives the mind into retrospection of some great works of ancient pioneers of the field of differential equations. As a teacher of differential equations, personal experience in the teaching of second order linear homogeneous differential equations, pioneered by Leonhard Euler [3], always points to students’ curiosity to understand why Euler’s assumption of the exponential function
(4)
fits well into the solution of the homogeneous linear differential equation of order 2, i.e.
(5)
with constant coefficients p and q. Both researchers and students often desire to explore the possibility of solving the Equation (3) above analytically to prove that
is the resultant solution rather than just intuitively assuming it as the solution and substituting it in order to prove its accuracy. Indeed, the view that
is an intuitive guess is widely shared by many mathematicians even though there has not been any better way of solving the problem [4]. For example, Nagy noted that the solution to Equation (3) is obtained by trial and error [4]. According to Nagy,
is first assumed to be the solution to equation (3) because the exponential cancels out of the equation leaving only a condition for “r”. Frankly, the fact that
proves to generates the solution to equation (3) only by trial and error has lead us to investigating these linear systems in order to determine their stability and consistency of their solutions.
Again, while studying the Euler linear homogeneous equation with variable coefficients
(6)
students are faced with similar questions as with the Equation (3). They often want to understand why the Euler substitution
(7)
fits well as the solution to the equation (6). Nagy also pointed that Equation (7) is obtained by assumption [4]. According to Nagy [4], the solution
is being sought for because it has the property that
and that
.
This clearly demonstrates that Euler’s solution were obtained by intuition rather than deduction. Any curious minded mathematician would want to derive the functions in Equations (4) and (7) in order to make assurance doubly sure. What can be made of this fact is that there does not seem to be adequate understanding of the deductive processes leading to the solutions to Equations (4) and (7). Indeed, this lack of adequate understanding must have lead Euler into the conclusion that the equation
(8)
has no solution in terms of elementary functions. In addition, the area of series solutions to homogeneous linear differential equations of order 2 with variable coefficients came into existence due to the perception that such equations cannot be solved in terms of known elementary functions or at least in terms of integral functions besides the Euler equation [5]. The general form of homogeneous linear ordinary differential equation of order 2 is given by
(9)
The conception that Equation (9) has no known solutions in terms of elementary functions or in terms integral functions besides the Euler equations has been held for centuries and perhaps this is the case because there has not been any better away to handle Equation (9). Even the series solutions are usually obtained by assumption [4] [5].
(10)
Note here that the intention is not to undermine assumption as an essential procedure in mathematical analysis. The point projected here is that assumptions actually make mathematics more presumptive and less deductive. It is important, therefore, to try the deductive procedures as well in order to provide sufficient ground for acceptance. It becomes pertinent to consider the possibility of proving the assumption in Equation (10) by some deductive method.
In the light of the foregoing, this paper presents a study that sets out to achieve the following objectives.
1) To prove that
is not just an intuitive assumption by some analytical derivation of it;
2) To prove that
is not just another ansatz but one that can be derived analytically; and
3) To prove that
is equally not just a guess but an analytically proven derivation, and
4) To show by these proofs that all linear homogeneous equations of order 2 actually derive the solutions by the same procedure irrespective of whether the coefficients are constants or variables.
2. Theoretical Framework
The framework upon which this study is based is quite simple, familiar and easy enough to comprehend. If we reexamined closely the actual meanings of the derivatives of the second order, third order, etc. then we might be well on the way to determining deductively the derivations of Equations (4), (7), (8) and (9) respectively.
Recall that
applying this logically, we can generate the proofs of statements (i), (ii), (iii) and (iv). The proofs will now be divided into the four sections following afterward.
2.1. Solution to the Second Order Linear Homogeneous Differential Equations with Constant Coefficient
Consider the second order linear homogeneous Equation (4) below
(5)
Using actual notations we have
Reinterpreting the notations, we get
Factorizing out
give us the following
It must be understood here that the expression in parenthesis is a differential operator with a constant. Thus we can denote it by D i.e.
(11)
The equation above becomes
(12)
From (12)
Integrating both sides we get
If we let
, then we have
(4)
As the solution to the Equation (5) we can equally determine the values of D and r by going back to Equation (11). Multiplying Equation (11) by y gives
But from (12)
Therefore it follows that by substitution
Equation (13) is called the characteristics equation of the differential operator D.
From (13)
Thus D has two values D, and D2 which implies also that r has two values
Therefore, the two linearly independent solutions of the Equation (5) are
(15)
We can equally find the characteristic equation in terms of r ? in fact Euler [5] already demonstrated that even though he was not quite aware that r depends on the differential operator D.
From (4),
,
and
which by substitution gives
(16)
From Equation (16),
(17)
Notice here that the values of r and D are different. They are, however, related by the formula
Another method we can use to find the solution to Equation (5) is by factorizing y out. See the procedure below.
Factoring out y gives us
Note here that
, thus if we let
then
By substituting
Since
, it follows that
(16)
This is the same characteristics equation that was obtained earlier, but
Multiplying both side by y, we get
(4)
Notice here too that the second method is much easier than the first
2.2. Solution to the Euler Second Order Linear Homogeneous Equation with Variable Coefficients
Given the Euler equation
(6)
using actual notations, we have
dividing through by x2,we get
Factoring out
as before, we get
Now let
(17)
Then by substitution, we get
(18)
From (17), multiplying by y gives
By substitution, we get
(19)
Equation (19) is the characteristic equation of the differential operator R(x).
From Equation (19),
(20)
By substituting (20) in (18), we get
Integrating both sides gives
Let
Then
(7)
Another method of doing this is to factor out y instead of
. Let us consider this method from (6), we get
Factoring out y gives
Let
Then
Since y ≠ 0, it follows that
(21)
From
But
Multiplying both sides by y gives
Let
, then
It is already common knowledge that
,
and
which by substitution leads to
(22)
Note here that Equation (22) is a very familiar characteristics equation and from it
This gives a different value of r when compared to the previous value.
3. Solution to the Equation
Euler’s prediction that Equation (7) would be an appropriate solution to Equation (6) was in indeed commendable achievement. However, it seemed likely that Euler did not envisage the possibility the Equation (8)
(8)
could equally have a solution in terms of elementary functions. While we cannot claim to have found a solution for Equation (8) with uttermost certainty, it is only fitting to demonstrate at this point that Equation (8) could possibly have a solution in terms of elementary functions. Rewriting Equation (8) with the usual notations, we have the following
Let
Then by substitution, we get
(23)
From (23), we have
Integrating both sides, we get
(24)
Recall also that
(25)
From both (23) and (25), we have
(26)
Equation (26) gives us the value of S(x) to establish the solution to Equation (8)
(27)
Substituting (27) in (24) gives
If we let
Then
(28)
Equation (28) is the solution to Equation (8). However, we cannot conclude that until we are able to establish more facts about it, so it is proper to take equation (28) as a tentative solution.
Another method we can use to find the solution to Equation (8) is given as shown below.
From Equation (8), we already have
If we decide to factor out y, then we have
Let
Then by substitution, we have
(29)
(30)
But
Multiplying both side by y, we get
Integrating both sides and ignoring constants, gives
If we let,
Then by substitution,
(28)
Now, in order to investigate Equation (28), we need to substitute it into the original differential equation. Thus from Equation (28) we have
Substituting into (8) gives
From which we get
(31)
And
(32)
From (32),
And from (31),
This value of m is same as that obtained earlier, therefore,
Since the solutions are independent, it follows that
Solution to the General Equation
It is generally agreed that an equation of the form
(33)
is called the general equation of second order homogenous linear differential equation. The solution to this equation can be found by adopting the same procedures as in the other cases. For instance, re-writing the equation, we get,
If we let
Then we get
(34)
But
Multiplying it by y gives
(35)
By comparing Equations (34) and (35), we can see that
(36)
From Equation (36), we get
And
If follows that from Equation (34)
(37)
By similar procedure, we can equally have
(38)
Now from Equation (34)
Integrating both sides gives us
(39)
Where
By substitution therefore,
Let
Then
(40)
Equation (40) may be an appropriate solution to the general Equation (33) it can be seen then that Equation (40) makes it possible for the solution to the general equation of the homogenous linear differential equation of order 2 to come in terms of an integral function.
Another way we can obtain a solution for Equation (33) is by factoring out y from (33)
Let
Then by substitution,
(41)
(42)
But
(43)
Substituting (42) in (43) gives
Let
Then by substitution
(40)
The Equation (40) works well if p and q are constants. It also works well if p and q are not constants in the Euler’s equation as well as the Equation (8). It can seen that for other homogenous equations with variable co-efficient, the workabilility of the Equation (40) depends on whether or not the function
and
the discriminant function
are both integrable since
may always be integrable, it is safe to say that if the discriminant function
is integrable, then the solution in Equation (40) is an
elementary function. If not then Equation (40) is not an elementary function. The case where the function
is not easily integrable leads us to think of possible power series solution. For simplicity, let
and
, then
But the series in the bracket converges to 1 if
Recall that
where
Let
Then
(44)
This implies that
(45)
From (45)
Thus, the Taylor series expansion of y at an ordinary point is given by
(46)
Equation (46) is therefore the general solution to equation (33)
4. Results and Discussion
The first objective of the study was to determine analytically the Equation (4) as the solution to the Equation (5) which is a homogenous linear differential equation with constant coefficients p and q. This was done through two different methods both of which yielded the same result
. Method 1 actually does something fascinating in a way it enables us to have a deeper view of the concept of characteristic equation. It enable us to understand that the r in the characteristic
Equation (16) is actually operator D where
. Method 2 does the same thing too but in this case the operator
.
The second objective of the study was to demonstrate that Equation (7) can be proven deductively to be the solution to Equation (6). By this, we mean that
is the solution to the Euler equation
. This was equally achieved through two different methods and the procedure is similar to the procedure in the first case.
In a very interesting way again, the third objective of this study was achieved by proving deductively that Equation (8) has a solution of the form in equation (28). By this we mean that
is the solution to the equation
. Notice that if this Equation (8) i.e.
is substituted into Equation (28) above, the resulting characteristic equation has two parts. One part is constant while the other part isobtained by the characteristic equation.
the constant part is obtained by the equation
thus making the complementary solution
to be of the form
. This makes the solution to appear like there are three independent solutions, whereas this is not the case. So what then does the constant stand for? It does in fact seem like there is more to be understood. To understand this, let us take advantage of the fact that the solutions are independent. Then we have to take
so that
And
Substituting the above into the left hand side of the original equation
We understand from here that the expression in the bracket
. However,
and
,
If this must be taken care of, then we must add an opposite function to make the equation balance. This opposite must as well be non-differentiable, which means that it is constant in a way. This non-differentiable function appears as part of the constant in the solution. There since
if follows that
is a non-differentiable independent function, this actually represent the non-differentiable part of the same function (8) which serves as the solution to Equation (8) i.e. Equation (28). This is largely accurate because such functions in the form
usually have both differentiable and non-differentiable regions. The diagram below illustrates this.
From the graphs in Figure 1 and Figure 2 above we can see that
has two parts, one part being differentiable while the other is non-differentiable i.e.
where
And
Thus for every function of the form
, the non-differentiable region is
. From the foregoing the solution to Equation (8) must therefore be of the form
Note that from the graph in Figure 1,
is differentiable within the region
, while it is non-differentiable outside this region since
is either 0 constantly or it has constant values at
and
. At these points, the function either has a derivative of 0 or its derivative is indeterminate, hence, it is non-differentiable.
The fourth objective of this paper was to demonstrate that Equation (10) is not just an ansatz to Equation (9) i.e.
is not just a guessed solution, the general second order linear homogenous differential equation
From this we learn that the function in Equation (40) is actually a good solution to the Equation (9) i.e.
where
is a solution to (9) if and only if the discriminant function
is integrable. In such a case the Equation (9) will have a solution in terms of elementary functions. However, if
is not integrable, then
must be converted to a binomial power
series which converges to
as thus leading to the function
So that
in this case the Taylor series expansion for this function becomes
Which reduces to
where
5. Conclusions
From the results obtained so far, we now draw the following conclusions:
1) The Euler’s exponential function
is indeed a deductive solution to the linear homogenous differential equation
.
where p and q = 0
2) The Euler’s equation
has the function
as its analytic solution and not as an ansatz.
3) The equation
has the function
as its solution such that the function has a differentiable and non-differentiable part.
4) The function
is indeed the solution to the general differential equation with variable coefficients
. This is not a guessed solution but an analytically derived solution to the equation above.