Modified Exponential Models

Calculus 2 · Axiom Academy

LESSON Modified Exponential Models One small change to decides everything: a constant inside settles to equilibrium, a constant outside grows without bound. 1. Two Ways to Modify Pure Growth Start from and ask: what if the rate doesn't depend on y alone? There are two natural edits. Subtract a fixed value inside the term — measuring distance from an equilibrium A — or add a fixed value outside it — a steady input b . Newton's Law of Cooling — the constant lives inside Growth with immigration — the constant lives outside The equation separates cleanly. Divide by y-A , integrate both sides, then exponentiate to undo the logarithm: Solving for y and naming the initial temperature y_0=y(0) gives the solution in equilibrium-plus-decay form: A constant floor. No matter the start, the solution is anchored to the ambient temperature. The initial gap, decaying away. With k<0 this term , so as . Worked instance — cooling coffee Coffee at C in a C room with cooling constant k=-0.1 per minute. Here A=20 and y_0-A=70 , so the solution is y(t)=20+70e^ -0.1t . At t=0 it reads ; as the exponential vanishes and the temperature settles to the room's . 3. Growth With Constant Immigration Now the other edit. A population grows naturally at rate ky but also gains a steady influx b — new arrivals, recruits, deposits. The two contributions simply add:

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