Mathematics and Statistics · Ch 8 — Differential Equation and Applications

Formation of a Differential Equation

A family of curves is described by an ordinary equation carrying one or more arbitrary constants (parameters). For example $y = mx$ is the family of all straight lines through the origin (one constant $m$ ), and $y = A e^{x} + B e^{-x}$ is a two-parameter family. Every such family satisfies a single differential equation, obtained by eliminating the arbitrary constants.

The rule. If a family contains $n$ arbitrary constants, differentiate the equation $n$ times and then eliminate all $n$ constants between the original equation and the derived ones. The result is a differential equation of order $n$ .

Worked illustration — one constant. Family $y = mx$ .

\frac{dy}{dx} = m.

Eliminate $m$ using the original ( $m = y/x$ ):

\frac{dy}{dx} = \frac{y}{x} \quad\Longrightarrow\quad x\frac{dy}{dx} = y.

One constant → order-1 differential equation.

Worked illustration — two constants. Family $y = A e^{x} + B e^{-x}$ .

\frac{dy}{dx} = A e^{x} - B e^{-x}, \qquad \frac{d^2y}{dx^2} = A e^{x} + B e^{-x}.

The second derivative equals the original $y$ , so the constants vanish at once:

\frac{d^2y}{dx^2} = y \quad\Longrightarrow\quad \frac{d^2y}{dx^2} - y = 0.

Two constants → order-2 differential equation.

Note

Number of Constants = Order of the Equation …

Definition 4Arbitrary constant

A parameter in the equation of a family of curves (e.g. $m$ in $y=mx$ , or $A,B$ in $y=Ae^x+Be^{-x}$ ) that varies from one member o …

Definition 5Formation of a differential equation

The process of eliminating the $n$ arbitrary constants from a family of curves by differentiating $n$ times, producing a differential equation of order $n$ sa …

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