Question

The solution of the differential equation \[\left( {xy - y} \right)dx + \left( {xy - x} \right)dy = 0\] is (where ‘C’ is the constant of integration)
A.\[xy = C{e^{{x^2} + {y^2}}}\]
B.\[{x^2}{y^2} = C{e^{{x^2} + {y^2}}}\]
C.\[C{x^2}{y^2} = {e^{{x^2} + {y^2}}}\]
D.\[xy = C{e^{x + y}}\]

Answer 1

Hint: We will use the variable separation method to solve this problem. We will use this method to solve this as equations which can be so expressed that the coefficient of \[dx\] is only a function of x and the coefficient of \[dy\] is only a function of y can be solved using the variable separation method and the given equation fits the criteria

Complete step-by-step answer:
We are given the differential equation \[\left( {xy - y} \right)dx + \left( {xy - x} \right)dy = 0\] and we have to find the solution of this equation.
We will first separate the variables x and y, that is, associate all the functions of x with \[dx\] and all the functions of y with \[dy\].
\[
  \left( {xy - y} \right)dx + \left( {xy - x} \right)dy = 0 \\
   \Rightarrow (xy - x)dy = - \left( {xy - y} \right)dx \\
   \Rightarrow x(y - 1)dy = - y\left( {x - 1} \right)dx \\
   \Rightarrow \dfrac{{y - 1}}{y}dy = - \dfrac{{x - 1}}{x}dx \\
\]
We now have the differential equation of the type \[f(x)dx = g(y)dy\]
We will now integrate both the sides and add a constant on either of the side.
\[
  \smallint \left( {\dfrac{{y - 1}}{y}} \right)dy = - \smallint \left( {\dfrac{{x - 1}}{x}} \right)dx \\
   \Rightarrow \smallint \left( {1 - \dfrac{1}{y}} \right)dy = \smallint \left( {\dfrac{1}{x} - 1} \right)dx \\
   \Rightarrow y - \log y = \log x - x + \log C \\
   \Rightarrow y + x = \log x + \log y + \log C \\
   \Rightarrow x + y = \log Cxy \\
   \Rightarrow {e^{x + y}} = Cxy \\
   \Rightarrow xy = C{e^{x + y}} \\
\]
Therefore, the solution of the differential equation is \[xy = C{e^{x + y}}\].
Thus, the answer is option D.

Note: It is necessary to add the arbitrary constant on one side otherwise the solution obtained will not be a general solution. Solving the differential equation means to give the general solution of the differential equation. Constant should be added only to one side of the differential equation. The constant of integration can be taken as c or log c depending upon the nature of the problem.