Question

Find the differential equation representing the family of curves $y=a{e}^{bx+5}$, where $a$ and $b$ are arbitrary constants.

Answer 1

Given:
The family of curve is $y=a{e}^{bx+5}$ .................. $i$
Differentiating the equation w.r.t. x,
$\frac{dy}{dx}=a.e^{bx+5}.b$
using equation $i$,
$\frac{dy}{dx}=y.b$ .............................$ii$
$\frac{1}{y}.\frac{dy}{dx}=b$ ..........................$iii$
Again differentiating equation $ii$ w.r.t. $x$,
$\frac{d^{2}y}{d{x}^2}=b.\frac{dy}{dx}=\frac{1}{y}.(\frac{dy}{dx}{)}^2$ [using $iii$]
$\therefore y.\frac{d^{2}y}{d{x}^2}=(\frac{dy}{dx}{)}^2$

Hence, the required differential equation is,

$y.\frac{d^{2}y}{d{x}^2}-{\left(\frac{dy}{dx}\right)}^2=0$