
The solution of \[ydx - xdy = 0\]
A) \[{y^2} = cx\]
B) \[y = c{x^2}\]
C) \[y = cx\]
D) \[{x^2} = cy\]
Answer
465.6k+ views
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.
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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