Question

What is the solution of the differential equation \[\dfrac{{dy}}{{dx}} = \dfrac{1}{x}\]?
A. \[y + \log x + c = 0\]
B. \[y = \log x + c\]
C. \[{y^{\log x}} + c = 0\]
D. None of these

Answer 1

Hint: Here, the first order differential equation is given. First, simplify the given equation by rearranging the terms or cross multiplication method. Then, integrate both sides of the equation with respect to the corresponding variables. In the end, solve the integrals using the standard integration formulas \[\int {ndx} = x + c\] and \[\int {\dfrac{1}{x}dx} = \log x + c\] to get the solution of the differential equation.

Formula Used: \[\int {ndx} = x + c\], where \[n\] is a constant.
\[\int {\dfrac{1}{x}dx} = \log x + c\]

Complete step by step solution: The given differential equation is \[\dfrac{{dy}}{{dx}} = \dfrac{1}{x}\].

Simplify the given equation.
\[dy = \dfrac{{dx}}{x}\]

Now integrate both sides with respect to the corresponding variables.
\[\int {dy} = \int {\dfrac{{dx}}{x}} \]
Apply the integration formulas \[\int {ndx} = x + c\], and \[\int {\dfrac{1}{x}dx} = \log x + c\].
We get,
\[y = log x + c\]
Therefore, the solution of the differential equation \[\dfrac{{dy}}{{dx}} = \dfrac{1}{x}\] is \[y = \log x + c\].


Option ‘B’ is correct

Note: Students often apply a wrong formula to integrate \[\dfrac {1}{x}\]. They integrate it by using the power formula of integration. But the correct formula is \[\int {\dfrac{1}{x}dx} = \log x + c\].
It is necessary to use an integration constant as soon as integration is performed if we solve a first-order differential equation by a variable method.