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Equilibrium in a linear equation

The equation $y' + ay = b$ has a fixed equilibrium $b/a$ , and the solutions approach it when $a$ is positive. The general solution is the equilibrium plus an exponential function that dies out. The initial value determines the constant. You find the equilibrium by setting the derivative equal to zero, so that $ay = b$ . Newton's law of cooling has the same form, with the room temperature as the equilibrium.

y = \frac{b}{a} + Ce^{-ax}

the solution

y \to \frac{b}{a}

the equilibrium

Symbols

$a$	the coefficient
$b$	the constant term
$C$	determined by the initial value

Example

$y' + 4y = 20$ has equilibrium $20/4 = 5$ . With $y(0) = 8$ , $y = 5 + 3e^{-4x}$ , and $y(0) = 8$ .

The equilibrium is a level, not the initial value. The initial value sits in the constant

C

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Part of Mathematics R2: Differential equations.