Question

What is the solution to the differential equation $\dfrac{{dy}}{{dx}} = \dfrac{y}{x}$ ?

Answer 1

Hint : This is a differential equation and to find its solution, we are going to first separate the terms and arrange the like terms on the same side of equal to sign and then integrate on both the sides. After integrating, using basic properties of logarithmic functions, we will get our solution.

Complete step by step solution:
The given differential equation is: $\dfrac{{dy}}{{dx}} = \dfrac{y}{x}$
Now, separate the variables and arrange like terms on the same side of equal to sign. So, the given differential equation can be written as
$\Rightarrow \dfrac{{dy}}{y} = \dfrac{{dx}}{x}$ - - - - - - - - - - (1)
By separating the variables we must keep in mind that dy and dx are in the numerators with their respective function.
Now, integrate on both the sides of equation (1), we get
$\Rightarrow \smallint \dfrac{1}{y}dy = \smallint \dfrac{1}{x}dx$ - - - - - - - - - - - (2)
Now, integration of $\dfrac{1}{x}$ is $\ln x$ and integration of $\dfrac{1}{y}$ is $\ln y$ .
So, equation (2) becomes
$\Rightarrow \ln y = \ln x + \ln c$ - - - - - - - - - - (3)
Now, using the general property of logarithmic functions,
$\log a + \log b = \log ab$
Equation (3) becomes,
$\Rightarrow \ln y = \ln xc$ - - - - - - - - - (4)
Removing $\ln$ from both sides of the equation, equation (4) becomes
$\Rightarrow y = xc$
Hence, the general solution to the differential equation $\dfrac{{dy}}{{dx}} = \dfrac{y}{x}$ is $y = xc$ .
So, the correct answer is “ $y = xc$ .”.

Note : Here, c is an arbitrary constant of integration and we can write $\ln c$ instead of c as our solution is in terms of natural logarithm. Writing c is very important in this question and one must not forget to add the constant at the end of integration.