Question

Solution of a differential equation \[xdy - ydx = 0\] represents
A. A rectangular hyperbola
B. Parabola whose vertex is at origin
C. Straight line passing through origin
D. A circle whose center is at origin

Answer 1

Hint:Given is the solution of a differential equation. We need to find that equation. So we will separate the variable or function with its respective derivative term.Then we will integrate the equation. After integrating we will definitely get the perfect answer.

Complete step by step answer:
Given is the solution of a differential equation,
\[xdy - ydx = 0\]
Separating the variables,
\[xdy = ydx\]
Now separating the respective variables,
\[\dfrac{{dy}}{y} = \dfrac{{dx}}{x}\]
Integrating both sides we get,
\[\int {\dfrac{{dy}}{y} = \int {\dfrac{{dx}}{x}} } \]
Taking the integral we get,
\[\log y = \log x + \log C\]
Removing the log on both the sides,
\[\therefore y = xC\]
This is the respective equation. Thus we know that is the equation of a straight line that passes through the origin.

Thus option C is the correct answer.

Note:In order to get the correct answer as we did the process above. But we should know that equation of all other options so that it becomes easy to tick the correct answer. Since all the geometrical shapes are having either vertex or center on the origin we should not miss a single step.
-Rectangular hyperbola \[xy = {C^2}\]
-Parabola with vertex at origin \[{y^2} = 4ax\]
-Circle with centre is at origin \[{x^2} + {y^2} = {r^2}\]