First slide

Introduction to Differential equations

Question

The curve such that at an point of it the square of the ordinate is twice the product of the x-intercept of the normal and the obscissa, is

Easy

A

circle
B

parabola
C

ellipse
D

hyperbola

Solution

$\begin{array}{l} Normal at P (x, y) is (Y - y) y_{1} + X - x = 0 \\ Its x -intercept = x + y \cdot y_{1} \\ 2 x (x + y \cdot y_{1}) = y^{2} \\ x (x + y \cdot y_{1}) = y^{2} \\ \Rightarrow 2 y y_{1} - \frac{y^{2}}{x} = - 2 x \\ \Rightarrow \frac{d (y^{2})}{d x} - \frac{y^{2}}{x} = - 2 x \\ I . F = \frac{1}{x} \\ Soln is \frac{y^{2}}{x} = \int - 2 x \cdot \frac{1}{x} d x \\ = - 2 x + c \\ \Rightarrow y^{2} + 2 x^{2} = c x \end{array}$

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