Differential equations describe how quantities change, and they show up in nearly every branch of science and engineering — this deck of 20 flashcards introduces the essential concepts for a first encounter with the topic. You'll learn what a differential equa...
Differential equations describe how quantities change, and they show up in nearly every branch of science and engineering — this deck of 20 flashcards introduces the essential concepts for a first encounter with the topic. You'll learn what a differential equation represents, the difference between a general and particular solution, and how to solve simple equations using separation of variables. The deck covers exponential growth and decay models, Newton's law of cooling, and the logistic growth equation, before introducing slope fields and Euler's method for approximating solutions numerically. It finishes with linear first-order equations and integrating factors. Suited to A-Level Further Maths and AP Calculus BC, this deck focuses on connecting each equation type to a real-world model — population growth, radioactive decay, cooling — so the abstract algebra has a concrete meaning you can visualise.
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An equation involving an unknown function and its derivatives, describing how a quantity changes in relation to another variable.
The highest derivative that appears in the equation — a first-order equation involves only the first derivative.
To find the function (or family of functions) that satisfies the equation for all values in its domain.
A general solution includes an arbitrary constant (representing a family of curves). A particular solution has a specific constant value, determined by an initial condition.
A differential equation paired with a specific condition (e.g.
𝑦
(
0
)
=
5
y(0)=5) used to determine the exact particular solution from the general solution.
Rearranging a differential equation so all terms involving
𝑦
y (and
𝑑
𝑦
dy) are on one side and all terms involving
𝑥
x (and
𝑑
𝑥
dx) are on the other, then integrating both sides.
𝑦
=
𝐶
𝑒
𝑘
𝑥
y=Ce
kx
, where
𝐶
C is a constant determined by initial conditions — the exponential growth/decay equation.
Exponential growth (k > 0) or decay (k < 0), such as population growth, radioactive decay, or compound interest.
𝑡
1
/
2
=
ln
2
𝑘
t
1/2
=
k
ln2
, the time for a quantity to reduce to half its initial value, where
𝑘
k is the decay constant.
A differential equation model stating that the rate of temperature change of an object is proportional to the difference between its temperature and the ambient temperature:
𝑑
𝑇
𝑑
𝑡
=
−
𝑘
(
𝑇
−
𝑇
𝑎
𝑚
𝑏
)
dt
dT
=−k(T−T
amb
).
A graphical representation showing the slope of solution curves at many points, generated directly from a differential equation without solving it explicitly.
Starting from an initial point, draw a curve that stays tangent to the small line segments at every point it passes through.
Logistic growth models a population that grows exponentially at first but levels off as it approaches a carrying capacity, unlike unrestricted exponential growth.
𝑑
𝑃
𝑑
𝑡
=
𝑘
𝑃
(
1
−
𝑃
𝑀
)
dt
dP
=kP(1−
M
P
), where
𝑀
M is the carrying capacity.
A constant solution to a differential equation where
𝑑
𝑦
𝑑
𝑥
=
0
dx
dy
=0 for all
𝑥
x, meaning the quantity doesn't change over time.
Numerically approximating solutions to differential equations by taking small steps along the slope field, useful when an exact analytical solution is difficult to find.
𝑦
𝑛
+
1
=
𝑦
𝑛
+
ℎ
⋅
𝑓
(
𝑥
𝑛
,
𝑦
𝑛
)
y
n+1
=y
n
+h⋅f(x
n
,y
n
), where
ℎ
h is the step size and
𝑓
(
𝑥
,
𝑦
)
=
𝑑
𝑦
/
𝑑
𝑥
f(x,y)=dy/dx.
An equation of the form
𝑑
𝑦
𝑑
𝑥
+
𝑃
(
𝑥
)
𝑦
=
𝑄
(
𝑥
)
dx
dy
+P(x)y=Q(x), solvable using an integrating factor.
A function, typically
𝜇
(
𝑥
)
=
𝑒
∫
𝑃
(
𝑥
)
𝑑
𝑥
μ(x)=e
∫P(x)dx
, multiplied through a linear differential equation to make the left side a recognisable derivative product, enabling direct integration.
Modelling population dynamics, radioactive decay, spread of disease (epidemiology), electrical circuits, and cooling/heating processes.
