2.1. The Core of the Problem of the Extraneous Solutions
Let us demonstrate the essence of the problem of the false solutions to irrational equations when using computer mathematical packages.
Consider an irrational equation with general parameters:
where
is the variable,
are real parameters, with
and
not equal to zero at the same time and
.
By solving symbolically Equation (1) using the standard tools of the mathematical package Mathcad, we obtain two roots (see the listing of the Mathcad program in
Figure 1).
We denote the found roots functions of several parameters as and .
Note that the formulas obtained for the roots
and
do not result in root variants for the case
, because for
Equation (1) has a single root due to the reduction of terms with the same coefficients (see the listing of the program part in Mathcad in
Figure 2).
Similarly, Equation (1) has only one root in the case or .
We will consider Equation (1) in the more general case of .
The analysis of the found roots and shows that they do not satisfy Equation (1) for all the parameter values. To demonstrate this, we introduce the quantities and which are the differences between the left and right sides of Equation (1) where the variable is substituted with the values of the found roots and respectively. Obviously, if the roots and do satisfy Equation (1), then the values and .
We plot
and
as functions of the parameter
for
(see
Figure 3a,b).
Figure 3a,b shows that the roots
and
do not satisfy Equation (1) for all values of the parameter
. Therefore, for
(
Figure 3a), the root
satisfies (1) only for
, and the root
satisfies for
. For
(
Figure 3b), the root
satisfies (1) only for
, while the root
does not satisfy at all (1).
Thus, using mathematical packages it is possible to ascertain the appearance of extraneous roots in the symbolic solution of an irrational equation of the form (1).
2.3. Application of the Proposed Method for Solving an Electrical Problem
Let us clarify the core of the problem of the extraneous solutions by an example of an applied electrical problem. It is requested to find the optimal distribution of the total power to two users of capacities and with a required total voltage less than the breakdown voltage. Users resistances are equal respectively to and .
According to the theory of electrical circuits, the expressions for the powers are described by the formulas [
12]:
Then, setting as parameters the resistances
and
, the total voltage
and the total power
, an irrational equation is derived for the sought variable
:
By solving symbolically Equation (8) using the standard tools of the mathematical package Mathcad, we obtain two roots (see the listing of the Mathcad program in
Figure 5).
It could be noted that by executing the calculations using the Mathematica or the Matlab packages, equivalent roots are obtained (see the listings of programs in Wolfram Mathematica [
13] and Matlab [
14,
15] packages in
Figure 6).
The found roots are functions of several parameters and .
After assigning to the parameters the following values
, the roots of Equation (8)
and
are plotted as function of the required total voltage
(
Figure 7).
Let us verify whether the functions
and
are actually solutions of Equation (8) for any value of
. In order to do this, we build a plot of the right side of Equation (8)—simply
, and a plot of the left side of Equation (8), by substituting in it the functions
and
instead of the sought variable
(
Figure 8). Obviously the solutions
and
are true solutions of the problem only for those values of
, where the plots of the left and right sides match.
According to the plot:
For U < 89.4 both solutions Pr1 (U, 8000, 1, 2) and Pr2 (U, 8000, 1, 2) are false;
For 89.4 ≤ U < 126.5 solution Pr1 (U, 8000, 1, 2) is false, while Pr2 (U, 8000, 1, 2) is true;
For U ≥ 126.5 both solutions Pr1 (U, 8000, 1, 2) and Pr2 (U, 8000, 1, 2) are true.
Thus, we have ascertained that the solution of Equation (8) shown in
Figure 5 includes extraneous roots.
2.3.1. Implementation of the Proposed Method
We use the proposed method for the elimination of the extraneous roots. We need to determine the inequalities that define the range of admissible values of the parameter—total voltage .
To determine the range of admissible values of the parameter
, it was proposed to use the properties of symmetric polynomials. Note that the Equation (8) here considered is not symmetric with respect to variables
. So the plots of the roots
and
are also not symmetric (
Figure 7). We introduce the notation:
Equation (8) becomes:
from this it derives:
Because the radicals must be non-negative, and .
Since that requirement is verified for .
From the condition
it follows the range of admissible values for the parameter
:
Comparing the plots of the roots in
Figure 8 and the plot of the function
in
Figure 9 for specific values of
, we notice that the lower bound for the value of
corresponds to the minimum admissible value of the variable
while the point of appearance of the second root
corresponds to the maximum admissible value
.
Finally, we solve the system given by Equation (8) and inequality (12):
The plot of the correct analytical solution obtained using Mathcad is shown in
Figure 10. Comparing it with the plot in
Figure 7, we see that the extraneous roots are eliminated, and, as noted earlier, in the interval
Equation (8) has only one root
, while in the interval
—two roots
and
.
2.3.2. A Special Case
Let us consider a special case of Equation (8), when the user resistances are equal
. In this case, the left side of Equation (8) is already initially a symmetric polynomial with respect to the variables
:
The plots of the solution (14), shown in
Figure 11 and similar to the plots in
Figure 7, have a symmetric appearance.
System (13) in this case reduces to the form:
The overall form of the analytical solution of system (15), obtained using Mathcad, is shown in
Figure 12. The analytical solution is written by using the conditional
if operator, which allows to determine a different type of solution for different ranges of values
.
The plot of the correct analytical solution of the system (15), shown in
Figure 13, has obviously a symmetric appearance, i.e., on the entire allowable interval
there are two symmetric roots
and
.
Thus, in this section, we have described a method for obtaining the correct analytical solution of an irrational equation using the properties of symmetry, demonstrating it using as example the solution of Equation (8).