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 curve,

$y=a{e}^{bx+5}$

or, $y=\left(a{e}^5\right)e^{bx}$

or, $y=c{e}^{bx}$ [ Putting $a{e}^5=c$]......(1).

Now differentiating both sides with respect to $x$ we get,

$\frac{dy}{dx}=cb{e}^{bx}$.

or, $b=\frac{1}{y}\frac{dy}{dx}$....(2). [ Using (1)]

Now aging differentiating both sides with respect to $x$ we get,

$\frac{d^{2}y}{d{x}^2}=c{b}^2{e}^{bx}$

or, $\frac{d^{2}y}{d{x}^2}=y\times \frac{1}{y^2}\times {\left(\frac{dy}{dx}\right)}^2$ [ Uisng (1) and (2)]

or, $\frac{d^{2}y}{d{x}^2}=\frac{1}{y}\times {\left(\frac{dy}{dx}\right)}^2$