JEE Main · 2021 · 26 Feb 2021 · Shift 1 · Differential Equations

The difference between degree and order of a differential equation that represents the family of curves given by

y^{2} = a (x + \frac{a}{2}), a > 0

is ______

Answer: 2

Solution

Differentiating

y^{2} = a (x + \frac{a}{2})

with respect to

x

gives

2 y \frac{d y}{d x} = a

.
Putting

a = 2 y \frac{d y}{d x}

back into the equation of the family,

y^{2} = 2 x y \frac{d y}{d x} + \frac{1}{2} (2 y \frac{d y}{d x})^{3/2}

and squaring to clear the half power,

(y^{2} - 2 x y \frac{d y}{d x})^{2} = \frac{1}{4} (2 y \frac{d y}{d x})^{3} = 2 y^{3} (\frac{d y}{d x})^{3}

Only the first derivative occurs, so the order is

1

, and its highest power is

3

, so the degree is

3

.
The difference between the degree and the order is

3 - 1 = 2

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