Newton’s law of cooling says an object’s temperature changes at a rate proportional to the difference between its temperature and the temperature around it: dT/dt = k(T − Tₛ). Solving this differential equation gives T = Tₛ + (T₀ − Tₛ)e^(kt), so the temperature moves toward the surrounding temperature Tₛ over time.
Exponential models with differential equations start from dy/dt = ky, which says the rate of change is proportional to the amount present. Separating the variables gives the solution y = y₀e^(kt), where y₀ is the starting amount. A positive k means exponential growth, and a negative k means exponential decay.
A separable differential equation is one you can write as dy/dx = g(x)·h(y), so all the y’s can move to one side and all the x’s to the other. To solve it, separate the variables, integrate both sides, add one constant C, and solve for y. An initial condition then gives the value of C.
Euler’s method is a step-by-step way to approximate the solution of a differential equation dy/dx = f(x, y) with a starting point. From the current point, you follow the tangent line for a short step h, land on a new point, find the new slope there, and repeat. Smaller steps give better estimates.
A slope field (or direction field) is a picture of a differential equation dy/dx = f(x, y). At many points (x, y), you draw a short line segment whose slope is f(x, y). Solution curves follow the segments, so a slope field shows what the solutions look like without solving the equation.
A differential equation is an equation that links a function with its derivatives, like y′ = 2y. To verify a solution of a differential equation, find the derivatives the equation uses, substitute them and the function into both sides, and simplify. If the two sides are equal for every x, the function is a solution.
The center of mass is the balance point of an object. For a flat region of uniform density between y = f(x) on top and y = g(x) below, it is the centroid: x̄ = (1/A) ∫ x(f − g) dx and ȳ = (1/A) ∫ ½(f² − g²) dx, where A is the region’s area.
Hydrostatic force is the total force a still liquid puts on a submerged surface, such as a dam or a window in a tank. Pressure grows with depth, so slice the plate into thin horizontal strips at depth h. Then F = ∫ ρ h w(h) dh, where ρ is the weight density and w(h) is the strip’s width.
Pumping liquid problems ask how much work it takes to pump liquid out of a tank. Slice the liquid into thin horizontal layers. Each layer’s work is its weight times the distance it must be lifted, so W = ∫ ρ A(y) (distance lifted) dy, where ρ is the liquid’s weight per unit volume and A(y) is the layer’s area.
The work done by a variable force F(x) moving an object along a line from x = a to x = b is W = ∫ F(x) dx. When the force is constant, work is just force times distance. When the force changes, you add up force times tiny distances, which is an integral.