AI Tutor
If you leave a boiling mug of coffee in a cool room, it drops 20 degrees in the first five minutes, but takes over an hour to drop the next 20 degrees.
The speed at which things change in the real world almost never stays constant β€” it continuously adapts based on the current state of the system.

The Translation Toolkit

A differential equation is any equation containing an unknown function and one or more of its derivatives.
Translating verbal rates begins with recognizing that the phrase 'rate of change of yy with respect to time tt' translates directly to the first derivative dydt\frac{dy}{dt}. If the quantity is decreasing or decaying, the rate must be negative (dydt=βˆ’k…\frac{dy}{dt} = -k \dots with k>0k > 0).
Under direct proportionality, a statement that a rate is 'proportional to the current amount yy' translates to dydt∝y\frac{dy}{dt} \propto y, giving dydt=ky\frac{dy}{dt} = ky for growth or dydt=βˆ’ky\frac{dy}{dt} = -ky for decay, where k>0k > 0 is the constant of proportionality.
How do we choose the correct sign when a quantity shrinks instead of expands?

Growth vs. Decay

In natural population growth, every individual contributes to future offspring, meaning the rate of growth dPdt\frac{dP}{dt} is directly proportional to the current population PP: dPdt=kP(k>0)\frac{dP}{dt} = kP \quad (k > 0)
For radioactive decay, unstable atoms disintegrate over time, so the mass MM is decreasing and requires an explicit negative sign: dMdt=βˆ’kM(k>0)\frac{dM}{dt} = -kM \quad (k > 0)
A common mistake is writing dMdt=kM\frac{dM}{dt} = kM for decay and assuming kk will automatically become negative later. In exam specifications, kk is always defined as a positive constant (k>0k > 0), so you must write the negative sign explicitly.