Lectures

Linear differential equations ( $1^{s t}$ order)

Linear y

no $y^{2}, \sqrt{y}, (y^{'})^{3}, \frac{1}{y}, \sin y, \dots$

a (x) y^{'} (x) + b (x) y (x) = c (x)

Homogenous linear equations

a (x) y^{'} (x) + b (x) y (x) = 0

a (x) \frac{d y}{d x} = - b (x) y (x)

\ln | y (x) | = \int \frac{d y}{d x} = - \int \frac{b (x)}{a (x)} d x = \dots + \ln C

y = C e^{- \int \frac{b (x)}{a (x)}}

Example

Typical notation

Method of variation of constants

Knowing that $y = C e^{- \int \frac{b (x)}{a (x)} d x}$