Form the differential equation of the family of curves $y\;=\;ae^{bx},$ where $a$ and $b$ are arbitrary constants.

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Form the differential equation of the family of curves $y\;=\;ae^{bx},$ where $a$ and $b$ are arbitrary constants.

Form the differential equation of the family of curves $y\;=\;ae^{bx},$ where $a$ and $b$ are arbitrary constants.

Equation of the family of curves, $y=a e^{b x}$ , where $a$ and $b$ are arbitrary constants.
Differentiating the above equation with respect to $x$ on both sides, we have,
$\begin{array}{l} y=a e^{b x}(1) \\ \frac{d y}{d x}=a b e^{b x}(2) \\ \frac{d^{2} y}{d x^{2}}=a b^{2} e^{b x} \end{array}$
$y \frac{d^{2} y}{d x^{2}}=a b^{2} e^{b x}\left(a e^{b x}\right)$ (Multiplying both sides of the equation by $y$ )
$y \frac{d^{2} y}{d x^{2}}=\left(a b e^{b x}\right)^{2}$ (Substituting equation 2 in this equation)
$y \frac{d^{2} y}{d x^{2}}=\left(\frac{d y}{d x}\right)^{2}$
This is the required differential equation.