[DEFINITIONS] First-order methods

Separable equation

A first-order ODE is separable when it can be written as

\frac{dy}{dx}=f(x)g(y).

On an interval where $g(y)\neq0$ , the variables can be placed on opposite sides. Zeros of $g$ must be checked before division because they may give constant solutions that the separated calculation does not contain.

[1]:Separation of variables

First-order linear equation

A first-order linear ODE in normal form is

y'+p(x)y=q(x).

The term $p(x)y$ couples $y$ to $y'$ , so direct integration of $y'$ and $q(x)$ separately does not solve the equation. An integrating factor makes the left side a product-rule derivative.

[1]:Linear ODE systems
[2]:Integrating factor

Exact differential equation

An equation in differential form

M(x,y)\,dx+N(x,y)\,dy=0

is exact on a domain when there is a potential function $F(x,y)$ such that

F_x=M,\qquad F_y=N.

Here $F_x$ means differentiate with respect to $x$ while holding $y$ fixed, and $F_y$ means differentiate with respect to $y$ while holding $x$ fixed. The later multivariable-calculus chapter develops this notation and the level-curve geometry in more detail.

Along a curve $y=y(x)$ , the differential equation becomes

M(x,y)+N(x,y)y'=0.

A potential turns this into the statement that $F$ is constant along the curve, as shown after the theorem below.

[1]:The exactness test and potential
[2]:Seen Questions

Homogeneous function

A function $F(x,y)$ is homogeneous of degree $k$ when

F(\lambda x,\lambda y)=\lambda^kF(x,y)

for nonzero scalars $\lambda$ wherever both sides are defined.

When the right side of $y'=F(x,y)$ is homogeneous of degree $0$ , scaling $(x,y)$ leaves it unchanged. On a suitable domain, it can therefore be expressed as a function of $y/x$ , which suggests the substitution $y=xv$ .

[1]:The substitution $y=xv$

Bernoulli equation

A Bernoulli equation has the form

y'+p(x)y=q(x)y^n, \qquad n\neq0,1.

The cases $n=0$ and $n=1$ are already first-order linear equations. For $n\neq0,1$ , the nonlinear power can be converted into a new dependent variable on a real branch where the required powers of $y$ are defined.

[1]:The Bernoulli substitution

Definitions from First-order methods