**Geometrical
Applications of Differential Equation**

**Geometrical Applications of Differential Equation: **We also use differential equations for finding the family of curves for which some conditions involving the derivatives are given. For this we proceed in the following way

Equation of the tangent at a point (x, y) to the curve y = f(x) is given by

Y – y = dy/dx (X – x)

At the X – axis, Y = 0

X = x – y/(dy/dx) (intercept on X – axis).

At the Y – axis, X = 0 and

Y = y – x(dy/dx) (intercept on Y – axis).

Similar information can be obtained for normal by writing its equation as

(Y – y) (dy/dx) + (X – x) = 0

**Example: **The slope of a curve, passing through (3, 4) at any point is the reciprocal of twice the ordinate of that point. Show that it is a parabola.

**Solution:**

Given condition dy/dx = 1/2y

2y. dy = 1. dx

Integration on both sides

∫2y. dy = ∫ 1. dx

2∫y. dy = ∫ 1. dx

2(y²/2) = x + c

The slope of a curve, passing through (3, 4)

Where x = 3 and y = 4

2 ((4) ²/2) = 3 + c

2(8) = 3 + c

16 = 3 + c

c = 16 – 3

c = 13

2(y²/2) = x + 13

y² = x + 13 which is parabola