Question

The solution of the D.E. xdy + ydx = 0 is:
$
  A.\;x + y = C \\
  B.\;xy = C \\
  C.\;\log (x + y) = C \\
  D.\;none\;of\;these \\
 $

Answer 1

Hint:To solve the given differential equation, we need to apply some methods with integration formulas. Here, in this problem first, we have to separate the terms with the same variables along the equal sign. Then next we have to apply the suitable integration formula, to eliminate the derivative terms. Its solution must not have any derivative term.

Complete step-by-step answer:
The given differential equation in the problem is:
xdy + ydx = 0
Now, separate the similar variable terms as,
xdy = -ydx
It can be further written as,
$\dfrac{{dy}}{y} = - \dfrac{{dx}}{x}$
Now, integrate both sides with respect to their variable, we get,
$\smallint \dfrac{{dy}}{y} = - \smallint \dfrac{{dx}}{x}$
$ \Rightarrow \smallint \dfrac{1}{y}dy = - \smallint \dfrac{1}{x}dx$….(1)
As we know that,
$\smallint \dfrac{{dx}}{x} = \ln \left| x \right| + $ Integration constant.
So, using this formula in equation (1) , we get
$\ln \left| y \right| + {c_1} = - \ln \left| x \right| + {c_2}$
Further we simplify it to get following result by using the logarithm formula $\log a + \log b = \log ab$
$
  \ln \left| y \right| + \ln \left| x \right| = {c_2} - {c_1} \\
   \Rightarrow \ln \left| {xy} \right| = C \\
 $ , where $C = {c_2} - {c_1}$ . C is integration constant.
Also, if the log value of some term is constant, then that term will also be constant.
So, we may write it as $\left| {xy} \right| = C$
Thus the solution of the differential equation will be $xy = C$ .

So, the correct answer is “Option B”.

Note:A differential equation is an equation which involves some ordinary derivatives of the variables and their functions. Solving a differential equation means to determine what function or functions will satisfy the given equation. Therefore, if we know what the derivative of a function is, then we can find the function itself. This is possible by using methods to get antiderivative. In other terms, we need to integrate the differential equation by using suitable integration formulas.