MATH528

MATH528 Lesson03: Linear ODEs, Bernoulli non-linear ODEs

Remark. Many mathematical techniques involve linear combinations

in linear algebra, $α 𝒖 + β 𝒗$ , $α, β \in ℂ$ scalars, $𝒖, 𝒗 \in ℂ^{n}$ , vectors
in calculus, $d (α f + β g) = α d f + β d g$ , $α, β \in ℝ$ scalars, $f, g : ℝ \to ℝ$ , functions

Definition. A first-order ODE of form $y^{'} + p (x) y = r (x)$ is said to be linear.

Definition. A first-order ODE of form $y^{'} + p (x) y = 0$ is said to be homogeneous linear.

Remark. A first-order homogeneous linear ODE is separable with solution

\frac{d y}{y} = - p d x \Rightarrow \ln y = - \int p (x) d x + \ln c \Rightarrow y = c e^{- \int p (x) d x}

Remark. A first-order non-homogeneous linear ODE leads to a differential form

(p (x) y - r (x)) d x + d y = 0, P (x, y) = p (x) y - r (x), Q (x, y) = 1,

for which

\frac{1}{Q (x, y)} (\frac{\partial P}{\partial y} - \frac{\partial Q}{\partial x}) = p (x),

such that

F (x) = \exp \int p (x) d x = \exp h,

is an integrating factor with property $F^{'} = p F$ such that the linear ODE can be rewritten in separable form

F y^{'} + F p y = F y^{'} + F^{'} y = \frac{d}{d x} (F y) = r