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

7.1

Modeling Situations with Differential Equations

Verifying Solutions for Differential Equations

Sketching Slope Fields

Reasoning Using Slope Fields

Approximating Solutions Using Euler’s MethodBC only

Finding General Solutions Using Separation of Variables

Finding Particular Solutions Using Initial Conditions and Separation of Variables

Exponential Models with Differential Equations

Logistic Models with Differential EquationsBC only

Lesson

Diagnostic

after diagnostic

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

where the uniqueness conditions hold (every AP equation), two distinct solutions cannot share a point, 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)

stepping FORWARD (h > 0): concave up UNDER-estimates, concave down OVER-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 POSITIVE NON-EQUILIBRIUM solution tends to M. The solution y = 0 stays at 0

Fastest growth (BC)

at M/2, where the curve INFLECTS. The rate → 0 near EITHER equilibrium, 0 and M

Logistic solved (BC)

M/(1 + Ae^(−kt)), with A = (M − P₀)/P₀

Unit 7 tools

Challenge bank

1 / 60

60 open-ended problems.

Read the question, work it out, then flip the card to compare your reasoning to the worked solution. Mark each card so you can return to the ones that still bite.

Question

Tap card to reveal explanation

Worked solution

Tap card to return to question

Cumulative assessment

Test the unit.

Twenty mixed items drawn from across all 9 topics, with guaranteed misconception-code coverage. Identifies which misconceptions still bite when you cannot see which topic the question came from. AB students are served only the AB items; BC students get the whole unit.

Begin assessment →

Check what stuck.

Units 1 through 7, drawn evenly so earlier units get the same share as this one. Twenty questions or a full 45-question section, your choice. Even coverage means this is a retention check rather than a score estimate. Follows the track you chose on the course page.

20 or 45questions

69topics

101codes covered

Begin assessment →