find the complete solution of p+q=1
Answers
Answer:
3.1 Introduction
Most of the problems encountered in scientific studies, when
formulated mathematically give rise to non-linear partial differential
equations i.e. the partial differential equations in which the partial
derivatives occur other than in the first degree. Here, we shall
consider only non-linear partial differential equations of order one.
3.2 Integrals of Partial Differential Equations of Order One
The most general form of a partial differential equation of
order one is f ( x , y ,z, p , q)=0, where x , y are the independent
variables, z is dependent variables and p≡
∂ z
∂ x
, q≡
∂ z
∂ y
are the
partial derivatives of order one.
The relation between the dependent variable and independent
variables obtained from the given partial differential equation is
called a solution or integral of the given partial differential
equation, provided the values of dependent variable and its partial
derivatives satisfy the partial differential equation.
The integrals of the partial differential equations of order one
involving independent variables x and y and dependent variable z
are generally classified as follows:
Given:
An equation p + q = 1.
To Find:
The solution set of the given equation.
Solution:
The given problem can be solved by using the concepts of linear equations in two variables.
1. When only a single equation is given, there is an infinite number of solutions possible.
2. Consider the equation p + q = 1. (p=1, q=0 and p=2, q=-1 and p=3, q=-2, etc. Infinitely many solutions are possible in this case).
3. When there are two linear equations the number of solutions can be zero, one, or infinitely many. But when there is only a single linear equation the solution set will be infinite.
4. The equation of the line p + q = 1 is a straight line that cuts the x-axis and the y-axis at one point. The range of the graph is from (-infinite, + infinite).
5. The Equation is also a one-one function and it is also an onto function. Hence, it is considered a bijection.
Therefore, the given equation has an infinite number of values as only a single equation is given.