Question

What is the solution of the differential equation \[\dfrac{{dx}}{x} + \dfrac{{dy}}{y} = 0\]?
A. \[xy = c\]
B. \[x + y = c\]
C. \[\log x\log y = c\]
D. \[{x^2} + {y^2} = c\]

Answer 1

Hint: All variables of the given differential equation are separated. Now we will take integration on both sides and apply the integration formula to get the solution of the given differential equation.

Formula Used: Integration formula of \[\dfrac{1}{x}\] is \[\log x + c\].
Product of logarithm formula:
\[\log a + \log b = \log ab\]

Complete step by step solution: Given differential equation is:
\[\dfrac{{dx}}{x} + \dfrac{{dy}}{y} = 0\]
Taking integration sign on both sides of the differential equation:
\[\int {\dfrac{{dx}}{x}} + \int {\dfrac{{dy}}{y}} = 0\]
Applying formula \[\int {\dfrac{1}{x}dx} = \log x + c\]
\[\log x + \log y = \log c\]
Applying product of logarithm formula:
\[\log xy = \log c\]
Simplify the above equation:
\[xy = c\]

Option ‘A’ is correct

Additional Information: There are two types of solutions of a differential equation. They are general solution and particular solution.
General solution: When we don’t know the exact value of the integration constant, then the solution of the differential equation is known as the general solution.
The general solution of differential equation represent the family of solution of the given differential equation.

Particular solution: When we know the exact value of the integration constant, then the solution of the differential equation is known as the particular solution.

Note: Students often do mistake to take integrating constant. Since we get the sum of two logarithm functions after integration, thus we will take \[\log c\] as an integrating constant instead of c as an integrating constant.