A differential equation relates a quantity to one or more derivatives. Translate proportional growth as a derivative proportional to the quantity; a rate depending only on time is a different model. To check a proposed solution, compute its derivative and substitute both function and derivative into the equation throughout the claimed domain. An initial condition picks a member of a solution family when uniqueness holds.
At each point (x,y) a slope field draws a short segment with slope given by the differential equation. If the rate depends only on y, slopes agree along horizontal lines. Zero slopes reveal equilibrium solution candidates, which must satisfy the equation. Trace solution curves tangent to the field; do not connect segment endpoints as if they were data values.
For a separable equation move the factors involving y to the dy side and the factors involving x to the dx side, then integrate. Keep an integration constant until applying the initial condition. Dividing by a function of y can discard equilibrium solutions, so inspect those separately. State the interval containing the initial point on which the expression solves the equation.
For a constant proportional rate, the solution is exponential. Determine the constant from data using logarithms and interpret its sign and units. Half-life and doubling time describe multiplicative changes, not fixed additive changes. A model in which a temperature difference decays uses that difference as the dependent quantity, producing a shifted exponential.
