Bernoulli equations — substitution: Foundations

This page assumes you have seen none of the notation. We build every symbol from the ground up, in the order they need each other, then hand you back to the parent Bernoulli topic.

0 · What is $y$ , and what is $x$ ?

Before any equation, agree on the actors.

Why the topic needs this: a differential equation is a statement about a curve we don't know yet. Everything below is a hunt for that curve.

1 · The derivative $\frac{d y}{d x}$ — steepness

Why two names? $\frac{d y}{d x}$ reminds you what is changing (the ratio of a small change in $y$ to a small change in $x$ ); $y^{'}$ is just shorthand when the meaning is clear. They are the same object.

Why the topic needs it: a Bernoulli equation contains $\frac{d y}{d x}$ . It tells us the slope at every point in terms of where we are — a recipe the curve must obey.

2 · What an ODE actually says

3 · Powers of $y$ : $y^{n}$ , $y^{- n}$ , $y^{1 - n}$

Everything special about Bernoulli lives in a power of $y$ . So nail powers.

Why the topic needs $y^{1 - n}$ specifically: the substitution names $v = y^{1 - n}$ . Where does $1 - n$ come from? Watch the arithmetic: to cancel the power $y^{n}$ from the equation we divide by it, turning $y$ into $y^{1 - n} = y^{1} / y^{n}$ (using the division rule above). That single subtraction $1 - n$ is the whole trick's fingerprint.

\frac{d y}{d x}

measure, in one word?

Slope (steepness) of the curve at a point.

Rewrite

y^{- 2}

and

y^{1/2}

without exponents.

\frac{1}{y ^{2}}

and

y

What is

y^{0}

, and why does that break the

n = 1

case?

y^{0} = 1

; with

n = 1

v = y^{1 - 1} = y^{0} = 1

is constant — useless.

P (x)

and

Q (x)

ever depend on

y

No — they depend on

x

only.

State the chain rule for

\frac{d v}{d x}

when

v

depends on

y

and

y

x

\frac{d v}{d x} = \frac{d v}{d y} \cdot \frac{d y}{d x}

Differentiate

v = y^{1 - n}

with respect to

x

\frac{d v}{d x} = (1 - n) y^{- n} \frac{d y}{d x}

What makes an ODE "linear in

y

y

and

y^{'}

appear only to the first power; no

y^{n}

, no products of

y

with

y^{'}

What does

\int \dots d x

do?

Finds the antiderivative — the function whose slope is the integrand (area accumulation).

Why does an arbitrary constant

C

appear after integrating?

Many curves share a slope field;

C

picks which one.

Connections

Bernoulli equations — substitution (index 4.6.5) (the parent — go here next)
Linear First-Order ODEs — Integrating Factor (the machine we reduce to)
Separable Equations
Exact Equations and Integrating Factors
Substitution Methods in ODEs
Riccati Equations

Foundations — Bernoulli equations — substitution

0 · What is $y$ , and what is $x$ ?

1 · The derivative $\frac{d y}{d x}$ — steepness

2 · What an ODE actually says

3 · Powers of $y$ : $y^{n}$ , $y^{- n}$ , $y^{1 - n}$

4 · Functions of $x$ only: $P (x)$ and $Q (x)$

5 · "Linear in $y$ " — the equation we wish we had

6 · The chain rule — the engine that makes $y^{n}$ vanish

7 · The integrating factor $μ (x) = e^{\int (1 - n) P d x}$ — one glimpse

8 · The arbitrary constant $C$

Prerequisite map

Equipment checklist

Connections