Question

The differential equation of the curve such that the initial ordinate of any tangent is equal to the corresponding subnormal is:
A. Linear
B. Homogenous
C. Exact
D. None of these

Answer 1

Hint: At a given point on a curve, the gradient of the curve is equal to the gradient of the tangent to the curve. Similarly, it also describes the gradient of a tangent to a curve at any point on the curve. To determine the equation of a tangent to a curve we have to find the derivative using the rules of differentiation.

Complete step by step solution: Equation of tangent (x,y) to curve y=f(x) is
Y−y=f′(x)(X−x)
Putting X=0, the initial ordinate of the tangent is y=xf′(x)
The subnormal at this point is given by \[y\dfrac{{dy}}{{dx}}\] ​, so we have
\[\begin{array}{l}
y\dfrac{{dy}}{{dx}} = y - x\dfrac{{dy}}{{dx}}\\
y\dfrac{{dy}}{{dx}} + x\dfrac{{dy}}{{dx}} = y\\
\dfrac{{dy}}{{dx}}(x + y) = y
\end{array}\]
When the variable or the number in multiplication changes the side it goes to the denominator, simply multiplication changes to division when it changes the sides.
\[\dfrac{{dy}}{{dx}} = \dfrac{y}{{x + y}}\]
Do cross-multiplication –
$\begin{array}{l}
dy(x + y) = y(dx)\\
x + y = y(\dfrac{{dx}}{{dy}})\\
\therefore \dfrac{{dx}}{{dy}} = \dfrac{{x + y}}{y}\\
\therefore \dfrac{{dx}}{{dy}} = \dfrac{x}{y} + \dfrac{y}{y}\\
\therefore \dfrac{{dx}}{{dy}} = \dfrac{x}{y} + 1\\
\dfrac{{dx}}{{dy}} - \dfrac{x}{y} - 1 = 0
\end{array}$ [Use concepts of the simplification of the equation ]
which is the linear equation
Hence, from the given multiple choices, option A is the correct answer.

Note: In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. To determine the equation of a tangent to a curve:
1. Find the derivative using the rules of differentiation.
2. Substitute the x - coordinate of the given point into the derivative to calculate the gradient of the
     tangent.
3. Substitute the gradient of the tangent and the coordinates of the given point into an appropriate form
     of the straight-line equation.
4. Make x the subject of the formula.
   The normal to the curve is the line perpendicular to the tangent of the curve at a given point.