CBSE Class 12 · 2018 · Differential EquationsVery Short Answer

Find the differential equation representing the family of curves

y = a e^{b x + 5}

, where

a

and

b

are arbitrary constants.

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

Solution

Two constants

(a, b)

require two differentiations.

y^{'} = ab e^{b x + 5} = b y

Differentiate

y^{'} = b y

again:

y^{''} = b y^{'}

Eliminate

b

using

b = \frac{y ^{'}}{y}

y^{''} = \frac{y ^{'}}{y} y^{'} = \frac{( y ^{'} ) ^{2}}{y} \Rightarrow y y^{''} = (y^{'})^{2}

CBSE Class 12 Board · 2018 · Subjective