Answer

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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.

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