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

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Last updated date: 13th Jun 2024
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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.