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 that $y=a{e}^{bx+5}....\left(i\right)$

$\therefore \frac{dy}{dx}=ab{e}^{bx+5}=yb$ (By (i))

$\Rightarrow \frac{y^{\prime }}{y}=b\Rightarrow \frac{yy"-y^{\prime }.y^{\prime }}{y^2}=0yy"-(y^{\prime }{)}^2=0$

Hence the required differential equation is $y\frac{d^{2}y}{d{x}^2}-{\left(\frac{dy}{dx}\right)}^2=0$