Question

The D.E. of the family of straight lines \[y = mx + \dfrac{a}{m}\] where m is the parameter is
A) \[x\dfrac{{dy}}{{dx}} = a\]
B) \[\left( {x - y} \right)\dfrac{{dy}}{{dx}} = a\]
C) \[x{\left( {\dfrac{{dy}}{{dx}}} \right)^2} - y\dfrac{{dy}}{{dx}} = a\]
D) \[x{\left( {\dfrac{{dy}}{{dx}}} \right)^2} - y\dfrac{{dy}}{{dx}} + a = 0\]

Answer 1

Hint: Here we will be differentiating the given solution of the equation and substitute the parameter in the given solution to find the corresponding differential equation.
Differential equation is an equation that relates the functions and their derivatives with the other.

Complete step by step answer:
The family of straight lines \[y = mx + \dfrac{a}{m}\] where m is the parameter has been provided. We are to find the corresponding D.E.
We know that in order to find the corresponding differential equation, we would need to differentiate the given solution of the equation and substitute the parameter in the given equation.

We know that the general equation for a family of straight lines is \[ax + by + c = 0\], which can be simplified into the slope intercept form \[y = mx + c\]. Hence, comparing the given equation \[y = mx + \dfrac{a}{m}\] with the slope intercept form, we understand that it is the equation of a family of straight lines.

Differentiating \[y = mx + \dfrac{a}{m}\], we get that \[\dfrac{{dy}}{{dx}} = m\].
So, by substituting \[\dfrac{{dy}}{{dx}} = m\] in \[y = mx + \dfrac{a}{m}\] we get that
\[y = \dfrac{{dy}}{{dx}}x + \dfrac{a}{{\dfrac{{dy}}{{dx}}}}\]
Now, we need to simplify this differential equation.
We would multiply both left hand side and right hand side by \[\dfrac{{dy}}{{dx}}\].
Hence, we get the equation as,
\[\begin{array}{l}
y\dfrac{{dy}}{{dx}} = {\left( {\dfrac{{dy}}{{dx}}} \right)^2}x + a\\
 \Rightarrow {\left( {\dfrac{{dy}}{{dx}}} \right)^2}x - y\dfrac{{dy}}{{dx}} + a = 0
\end{array}\]
Thus, the required differential equation of the family of straight lines \[y = mx + \dfrac{a}{m}\] where m is the parameter is
\[{\left( {\dfrac{{dy}}{{dx}}} \right)^2}x - y\dfrac{{dy}}{{dx}} + a = 0\].

Note: Differential equation of a given system of curves is formed by eliminating the variable parameters of the curve by the differential of the dependent variable with respect to the independent variable.