Question

The differential equation of all straight lines passing through the origin is
A. \[y = \sqrt {x\frac{{dy}}{{dx}}} \]
B. \[\frac{{dy}}{{dx}} = y + x\]
C. \[\frac{{dy}}{{dx}} = \frac{y}{x}\]
D. None of these

Answer 1

Hint:
First, use the formula \[(y - {y_1}) = m(x - {x_1})\] to write the equation of the line passing through the origin. Then, with respect to \[x\], differentiate both sides, and finally, substitute the value of \[m\] in it. The differential equation of the family of all straight lines passing through the origin must be found here. We know that in order to find the differential equation of a family of curves, we must first find the equation of the curve.
Formula use:
Equation of line passing through the point \[({x_1},{y_1})\]
\[(y - {y_1}) = m(x - {x_1})\]
\[y = {x^n}\]
Differentiate with respect to x
\[\frac{{dy}}{{dx}} = n{x^{n-1}}\]

Complete step-by-step solution:
We know that the equation of the line with slope\[m\]and the point \[({x_1},{y_1})\] which it passes through is\[(y - {y_1}) = m(x - {x_1})\]
So, by substituting \[{y_1} = 0\] and \[{x_1} = 0\] in the above equation, we get the equation of the line passing through the origin as,
\[(y - 0) = m(x - 0)\]
This can be also written as,
\[y = mx \ldots ....(1)\]
Now, we divide the either side of the equation (1) by m:
\[\frac{y}{x} = m.........(2)\]
We know that, according to product rule of differentiation:
Since, \[\frac{d}{{dx}}(f.g) = g\left( {\frac{{df}}{{dx}}} \right) + f.\left( {\frac{{dg}}{{dx}}} \right)\]
Now, by differentiating both sides of equation (1) with respect to \[x\], we get
\[\frac{d}{{dx}} = m\left( {\frac{{dx}}{{dy}}} \right) + x\left( {\frac{{dm}}{{dx}}} \right)\]
For a particular value of x and y, we know that\[m\]is constant.
So, \[\left( {\frac{{dm}}{{dx}}} \right) = 0\].
Thus, we obtain
\[\frac{{dy}}{{dx}} = m(1) + x(0)\]
Now, by substituting the value of \[m\] from equation (2), we obtain
\[\frac{{dy}}{{dx}} = \left( {\frac{y}{x}} \right)\]
We have to subtract \[\frac{y}{x}\] from both sides of the above equation, we obtain
\[\frac{{dy}}{{dx}} - \left( {\frac{y}{x}} \right) = 0\]
Rewrite the above equation explicitly by having all the terms on one side:
\[\frac{{dy}}{{dx}} = \left( {\frac{y}{x}} \right)\]
Therefore, the differential equation of all straight lines passing through the origin is \[\frac{{dy}}{{dx}} = \left( {\frac{y}{x}} \right)\]
Hence, the option C is correct.
Note:
Students must remember that in order to find the differential equation of any curve, they must remove all constants from the equation, just as we did in the above solution. Students can also test this differential equation by changing the value of \[\frac{{dy}}{{dx}}\] in the differential equation and determining whether or not the original equation of the curve is obtained.