Differential Equations: every key term you need (+ practice quiz)
30 flashcard terms for AP Calculus BC Unit 8, written to match the course framework. Read them here, drill them as flashcards, or take the 27-question quiz. Free, no account needed.
General: includes constant C (all solutions). Particular: specific solution from initial condition.
Homogeneous vs Nonhomogeneous
Homogeneous: dy/dx = f(y) (no explicit x term). Nonhomogeneous: dy/dx = f(x,y) (includes x).
Unit 8 Summary
Differential equations model rates of change. Separation of variables solves many. Exponential/logistic growth fundamental applications.
Order and Solution of an ODE
The order is the highest derivative present; a solution is any function that satisfies the equation identically on an interval.
General vs Particular Solution
The general solution carries an arbitrary constant; an initial condition pins the constant down and yields the particular solution.
Slope Fields
At each grid point the segment has slope dy/dx computed from the equation; solution curves follow the segments without crossing one another.
Reading a Slope Field
Horizontal segments occur where dy/dx=0, revealing equilibrium solutions; segments identical along a horizontal line mean the equation depends only on y.
Separation of Variables
Rewrite as g(y)dy = f(x)dx, integrate both sides, add a single constant, then solve for y and apply the initial condition.
Losing Solutions
Dividing by a factor like y can discard the constant solution y=0; check whether such equilibria satisfy the original equation.
Exponential Model
dy/dt = ky gives y = y0·e^(kt): growth for k>0, decay for k<0, with a constant doubling or half-life time.
Newton's Law of Cooling
dT/dt = -k(T - T_env) has solution T = T_env + (T0 - T_env)e^(-kt); the object approaches ambient temperature asymptotically.
dP/dt = kP(1 - P/L) grows nearly exponentially at first and levels off at the carrying capacity L.
Logistic Inflection Point
The population grows fastest at P = L/2, where the solution curve changes concavity; this is a standard AP question.
Logistic Solution Form
P(t) = L/(1 + Ae^(-kt)) with A determined by the initial population. As t→∞, P→L regardless of the starting value.
Euler's Method
y_(n+1) = y_n + h·f(x_n, y_n) steps along tangent lines. Smaller steps reduce error but the method remains only first-order accurate.
Euler Error Direction
On a concave-up solution, tangent lines lie below the curve, so Euler's method underestimates; concave-down solutions are overestimated.
Equilibrium Solutions
Constant solutions occur where dy/dt = 0; they are stable if nearby solutions approach them and unstable if solutions move away.
Modeling Rate Statements
'Proportional to' becomes k times the quantity; 'proportional to the difference' becomes k times a subtraction, as in cooling problems.
Domain of a Solution
A particular solution is valid only on the largest interval containing the initial x on which the solution is differentiable and continuous.
Sign Analysis of dy/dx
Determining where dy/dx is positive or negative predicts whether the solution rises or falls without solving the equation.
Second Derivative from the ODE
Differentiate dy/dx = f(x,y) implicitly to obtain d²y/dx², which gives concavity and justifies Euler over- or under-estimation.
Mixing Problems
Rate of change of solute equals inflow concentration times inflow rate minus outflow concentration times outflow rate, giving a first-order linear model.
Verifying a Proposed Solution
Substitute the candidate and its derivative into the equation; matching both sides identically confirms it, and the initial condition confirms the constant.