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Unit 7
5–10% of examA differential equation describes how something changes, like a population that grows faster when it's bigger. In this unit you'll write differential equations from words, picture their solutions with slope fields, and solve the separable ones exactly. You'll also study exponential growth and decay, and BC students add Euler's method and logistic growth.
Longer videos that cover the whole unit. Good for a first pass or a final review.
A differential equation is an equation that includes a derivative, like dy/dt = ky. You'll translate sentences into these equations: for example, "the rate of change of y is proportional to y" becomes dy/dt = ky, where k is a constant.
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A solution to a differential equation is a function, not a number. To check one, take the needed derivatives, substitute into the equation, and see if both sides match. Most differential equations have a whole family of solutions, not just one.
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A slope field is a grid of short line segments. At each point, the segment's slope is the value of dy/dx that the differential equation gives at that point. To sketch one by hand, plug each grid point into the equation and draw a small segment with that slope.
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You can use a slope field to sketch the solution curve through a given point by following the segments, and to match a slope field with its differential equation by checking where slopes are zero, positive or negative. Because a differential equation has a whole family of solutions, different starting points give different curves.
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Euler's method estimates a solution step by step: from a known point, follow the tangent line for a small step Δx, using new y = old y + (dy/dx at the old point) × Δx, then repeat from the new point. Smaller steps usually give better estimates.
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If a differential equation can be rewritten with all the y's and dy on one side and all the x's and dx on the other, integrate both sides to solve it. Add the constant C to get the general solution, which describes a whole family of curves.
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An initial condition, like y(0) = 3, lets you solve for C and get the particular solution, which is the one curve through that point. Do this right after integrating, and watch for domain restrictions, such as values where the solution isn't defined.
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When a quantity changes at a rate proportional to its size, dy/dt = ky, and the solution is y = y₀eᵏᵗ, where y₀ is the starting amount. If k > 0 you get exponential growth, and if k < 0 you get exponential decay.
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A logistic model, dP/dt = kP(1 − P/L), describes growth that slows down as it gets close to a limit L called the carrying capacity. For a positive starting value, P approaches L over time, and the growth rate dP/dt is greatest when P = L/2, half the carrying capacity.
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