Differential Equations
Nine topics on equations that describe a rate. Modeling a sentence into a derivative, verifying that a candidate function really solves one, slope fields to sketch and to reason from, the equilibrium solutions nothing crosses, separation of variables for the general family and the particular member an initial condition selects, the domain that member actually lives on, exponential and cooling models, and, for BC, Euler's method and logistic growth.
AB exam 5-10%BC exam 5-10%9 topics
Topics
Key forms For every problem in this unit
Rate language
becomes a DERIVATIVE, never the quantity itself
Proportional to
k times WHATEVER FOLLOWS, parentheses included
Inversely proportional
k OVER it, not k times it
The constant k
units of 1/time: not an amount, not a duration
A solution
a FUNCTION; check BOTH sides, for every input
Slope field
substitute the point; draw that slope there
Which variable
x only: columns match. y only: rows match
Zero segments
solve f(x, y) = 0; a horizontal line is an EQUILIBRIUM
Solution curves
never touch, so nothing crosses an equilibrium
Euler (BC)
y + h · f(x, y), with the slope RECOMPUTED each step
Step size (BC)
h = (b − a)/n, and NEGATIVE going left
Euler error (BC)
concave up UNDER-estimates; it accumulates
Separable
the right side must FACTOR into x stuff times y stuff
Separate fully
a stray y on the dx side means you cannot integrate yet
One constant
added at the INTEGRATION step, on either side
Exponentiating
e^C becomes a FACTOR: y = Ce^(kx), never e^(kx) + C
Dividing by y
check the constant solution it discards
Particular solution
use the point; a family is not an answer
Condition off zero
the exponential does NOT cancel
Which branch
the sign of the given value picks it, once
Domain
largest CONNECTED interval holding the initial point
Exponential model
y = y₀e^(kt); doubling and half-life are ln 2 over |k|
Newton cooling
the DIFFERENCE decays: T = Tₐ + Ce^(kt)
Logistic (BC)
equilibria at 0 and M; every solution tends to M
Fastest growth (BC)
at M/2, where the curve INFLECTS. Slowest near M
Logistic solved (BC)
M/(1 + Ae^(−kt)), with A = (M − P₀)/P₀