This is the other half of 7.1: most differential equations begin life as a description. The trick is to spot two things — the quantity whose rate of change is being described (that's a derivative) and what that rate is proportional to. Pick a situation, read it, then reveal how each phrase turns into a symbol.
On the exam, “proportional to” almost always means times a constant k, and a word like decays, cools, or shrinks means that constant carries a minus sign.
A function solves a differential equation when its slope matches what the equation demands — at every point. Pick a candidate, make your prediction, then sweep across: at each point the teal tick is the slope the equation wants, and you can watch whether the curve actually has that slope.
Lock in verifying first: to check a proposed solution you never have to solve anything — just differentiate it and see whether it satisfies the equation.
A differential equation almost never has a single solution. It has a family — one curve for each value of the constant C. An initial condition, a single point the solution must pass through, selects exactly one. Click anywhere on the plot to drop an initial condition and watch the family collapse to one particular solution.
One differential equation, infinitely many solution curves. The initial condition is what turns a general solution (the one with a +C) into the single particular solution a real problem is asking for.
An ordinary algebra equation hunts for a number. A differential equation hunts for a function — one whose derivative fits a given rule.
y(x) is a solution exactly when substituting it (and its derivative) makes the equation true for all x. Verifying never requires solving.
Differentiation erases constants, so solutions arrive in families. An initial condition like y(0) = 2 selects the single curve you want.
Most DEs begin as a sentence: "the rate of change is proportional to the amount." Translate that into symbols and you've modeled it.
Six quick checks across both skills — translating a situation into an equation, and verifying a proposed solution. Try each one before you reveal the worked answer.
Six fast multiple-choice checks across the whole lesson. Pick an answer to see whether it's right and why — then work through all six.
An antiderivative-type equation has a whole family of solutions. Leave off the +C and you've written one curve where the general solution needs all of them — and you'll lose the point on a free-response.
“Decays,” “cools,” and “shrinks” all mean the rate is negative. dT/dt = k(T − A) describes heating; cooling needs the minus sign: dT/dt = −k(T − A).
If the quantity is a population P over time t, the rate is dP/dt — not dy/dx. Match the letters in the equation to the quantities in the problem.
Without an initial condition, a differential equation has infinitely many solutions. You need a point the curve passes through to pin down the single particular solution.
To check a proposed solution you never integrate. Differentiate the candidate, substitute into the equation, and see whether both sides agree — it's a 30-second job.
“Proportional to the amount” is not dy/dt = y. It's dy/dt = ky — proportionality always brings a constant along with it.