CBSE Class 12 · 2005 · Differential EquationsVery Short Answer

Form the differential equation representing the family of curves

y^{2} - 2 a y + x^{2} = a^{2}

, where

a

is an arbitrary constant.

Answer: $(x^{2} - 2 y^{2}) (\frac{d y}{d x})^{2} - 4 x y \frac{d y}{d x} - x^{2} = 0$

Solution

There is one arbitrary constant

a

, so we differentiate once and eliminate

a

.
Differentiate

y^{2} - 2 a y + x^{2} = a^{2}

w.r.t.

x

2 y y^{'} - 2 a y^{'} + 2 x = 0 \Rightarrow a = \frac{y y ^{'} + x}{y ^{'}}

Substitute this

a

back into the given relation

y^{2} + x^{2} - 2 a y = a^{2}

. Multiplying throughout by

(y^{'})^{2}

and simplifying:

y^{2} (y^{'})^{2} + x^{2} (y^{'})^{2} - 2 y (y y^{'} + x) y^{'} - (y y^{'} + x)^{2} = 0

(x^{2} - 2 y^{2}) (y^{'})^{2} - 4 x y y^{'} - x^{2} = 0

CBSE Class 12 Board · 2005 · Subjective