Question

The differential equation obtained by eliminating arbitrary constants from \( bx + ay = ab \) is \( \frac{d^2y}{dx^2} = 0 \).

Hint:

To form a differential equation from a relation with \(n\) arbitrary constants, you generally need to differentiate the relation \(n\) times and then eliminate the constants from the resulting equations.

Answer 1

Step 1: Differentiate the given equation \( bx + ay = ab \) with respect to \( x \). \[ b + a\frac{dy}{dx} = 0 \] Step 2: This first derivative shows that \( \frac{dy}{dx} = -\frac{b}{a} \), which is a constant value since \(a\) and \(b\) are constants.
Step 3: Differentiate the result from Step 1 again with respect to \( x \). The derivative of a constant (\(b\)) is zero, and the derivative of a constant times a function (\(a\frac{dy}{dx}\)) is the constant times the derivative of the function. \[ 0 + a\frac{d^2y}{dx^2} = 0 \implies \frac{d^2y}{dx^2} = 0 \] The statement is true.