Question

Solution of the differential equation $xdy - ydx = 0$ represents
\[\left( a \right){\text{ }}a{\text{ }}rectangular{\text{ }}hyperbola\]
\[\left( b \right){\text{ }}parabola{\text{ }}whose{\text{ }}vertex{\text{ }}is{\text{ }}at{\text{ }}origin\]
\[\left( c \right){\text{ }}straight{\text{ }}line{\text{ }}passing{\text{ }}through{\text{ }}origin\]
\[\left( d \right){\text{ }}a{\text{ }}circle{\text{ }}whose{\text{ }}centre{\text{ }}is{\text{ }}at{\text{ }}origin\]

Answer 1

Hint:
For this question, we will first equate the equation and then we will keep the same term at one side and then taking log both sides and we will get the equation, which we have to find out from the four options given to us.

Complete step by step solution:
 The equation given to us is $xdy - ydx = 0$
On equating the equation, we get
$ \Rightarrow xdy = ydx$
Now taking the same term at one side, we get
$ \Rightarrow \dfrac{{dy}}{y} = \dfrac{{dx}}{x}$
So now on taking the integral both the sides, we get
$ \Rightarrow \int {\dfrac{{dy}}{y}} = \int {\dfrac{{dx}}{x}} $
On doing the integration, we get
$ \Rightarrow \log y = \log x + \log c$
Now by using the properties of the log, as we know when two logs are added then it comes under the properties of log multiplication.
That is mathematically it can be written as
$ \Rightarrow \log y = \log c.x$
Since both sides have log common so it will cancel out, then we get
$ \Rightarrow y = c.x$, which is the equation of a straight line.
Hence, the equation represents the straight line.
Therefore, the option $\left( c \right)$is correct.

Additional information:
Rectangular hyperbola means that the asymptotes of the hyperbola are perpendicular lines. It is a bend with one hub of balance. On the off chance that it was a mirror, light beams corresponding to the axis would be reflected through a typical point called the focus. The bend can be acquired by cutting a cone with a plane corresponding to one of its generators.

Note:
To answer this type of question, we should know integration and differentiation and apart from this we should also know the equation of each of the planes. By putting the random number we can also know the path of the equation if we have the equation given to us.