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# The solution of $ydx - xdy = 0$ A) ${y^2} = cx$ B) $y = c{x^2}$ C) $y = cx$ D) ${x^2} = cy$

Last updated date: 13th Jun 2024
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Hint:
Here we have to solve the given differential equation. For that, we will equate the terms and divide the terms in such a way that the equation comes in $\dfrac{{dy}}{{dx}}$ form. Then we will use integration to solve the equation. On further simplification, we will get the solution of the differential equation.

Complete step by step solution:
The given differential equation is $ydx - xdy = 0$.
Taking the term $xdy$ to right side of equation, we get
$ydx = xdy$
Now, we will divide all the terms on both sides of the equation by the term $xy$.
$\Rightarrow \dfrac{{ydx}}{{xy}} = \dfrac{{xdy}}{{xy}}$
On further simplification, we get
$\Rightarrow \dfrac{1}{x}dx = \dfrac{1}{y}dy$
Now, we will integrate both the terms.
$\Rightarrow \int {\dfrac{1}{x}dx} = \int {\dfrac{1}{y}dy}$
On integrating the terms, we get
$\log x = \log y + \log C$
We have added constant $\log C$ because it is an indefinite integral.
We know by the property of logarithmic function $\log a + \log b = \log ab$.
Now, we will be using the same property of logarithmic function for the term $\log y + \log c$.
Thus, the above equation becomes;
$\Rightarrow \log x = \log yC$
Rewriting the equation, we get
$\Rightarrow x = yC$
Dividing $C$ on both the side, we get
$\Rightarrow \dfrac{1}{C}x = y$
As $\dfrac{1}{C}$ is also a constant we can denote it as $c$.
Thus, the final equation becomes;
$\Rightarrow y = cx$

Hence, the correct answer is option C.

Note:
Here, we need to keep basic integration property in mind. A logarithmic function is defined as a function, which is inverse of the exponential function.
Some important properties of logarithmic function are:-
The logarithm of a product of two or more terms is equal to the sum of the logarithm of each term.
The logarithm of a division of two terms is equal to the difference of the logarithm of these two terms.