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The differential equation obtained by eliminating arbitrary constants from the expression \(y = ae^{bx}\) is :

  • A.\(y\dfrac{d^{2}y}{dx^{2}} - \left(\dfrac{dy}{dx}\right)^{2} = 0\)
  • B.\(y\dfrac{d^{2}y}{dx^{2}} - x\dfrac{dy}{dx} = 0\)
  • C.\(\dfrac{d^{2}y}{dx^{2}} + xy\left(\dfrac{dy}{dx}\right)^{2} = 0\)
  • D.\(\dfrac{d^{2}y}{dx^{2}} - xy\dfrac{dy}{dx} = 0\)

Answer: A

The equation is \(y\dfrac{d^{2}y}{dx^{2}} - \left(\dfrac{dy}{dx}\right)^{2} = 0\).

From \(y = ae^{bx}\): \(y' = by\) and \(y'' = b^{2}y\). Eliminating \(b\), \(b = y'/y\), so

\[ \begin{aligned} y'' &= \frac{(y')^{2}}{y}, \\ \text{that is,} \quad yy'' - (y')^{2} &= 0 \end{aligned} \]

Both constants disappear, as required for a family with two parameters and a second-order equation.

The other options contain \(x\) explicitly, but the family \(ae^{bx}\) is unchanged by shifts in \(x\) (a shift only changes \(a\)), so its differential equation cannot involve \(x\).

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