Question

Find the differential equation of the family of curves $ {y^2} = 4ax $

Answer 1

Hint:
This type of question is very easy and easily we can solve and the only concept that is needed is differentiation. So in this question, we will differentiate the equation with respect to $ x $ and then put the values in place of the equation and we will get the differential equation.

Complete step by step solution:
 we have the curve whose equation is $ {y^2} = 4ax $ , let’s name it equation $ 1 $
Now we have to find out the differential equation for this we will differentiate the above equation both the sides with respect to $ x $ , we get
\[ \Rightarrow 2y\dfrac{{dy}}{{dx}} = 4a\], let’s name it equation $ 2 $
So from equations $ 1 $ and $ 2 $ , we get
 $ \Rightarrow {y^2} = 2y\dfrac{{dy}}{{dx}}x $
So on solving more, we get
\[ \Rightarrow 2x\dfrac{{dy}}{{dx}} = y\]
So the differential equation can be written as
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{y}{{2x}}\]
Now taking all the terms in one side, we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} - \dfrac{y}{{2x}} = 0\]

Therefore \[\dfrac{{dy}}{{dx}} - \dfrac{y}{{2x}} = 0\], will be the differential equation.

Additional information:
An answer to a differential condition is a connection between the factors (autonomous and subordinate), which is liberated from subsidiaries of any request, and which fulfills the differential condition indistinguishably. Presently we should dive into the subtleties of what 'differential conditions arrangements' are!
These conditions have a stunning ability to figure out our general surroundings. Also, they are utilized in a wide assortment of orders, from science, financial matters, science, material science, and designing. Besides, they can characterize outstanding development and rot, the populace development of species, or the adjustment in speculation return after some time.

Note:
We have seen that for solving and differentiating these types of equations we only need to memorize some easy differential formulas and through it, we can solve them easily. So with practice, it will be clearer when and where to use the formula.