🌱 Digital Garden

First order first degree

Many be written as #

$$ \begin{align*} g(y)y’&= f(x)\\ g(y)dy &= f(x)dx\\ \int g(y)dy &= \int f(x)dx +C \end{align*} $$

These are called separable ODEs

Applications in Mathematical Modelling

Homogenous ODE #

$$ \begin{align*} y’=f(y/x)\\ Let\\ y=ux:y’=u’x+u\\ So,\\ u’x+u=f(u)\\ u’x=f(u)-u\\\ \frac{du}{f(u)-u}=\frac{dx}{x} \end{align*} $$

We made it separable

More general way #

$$ \begin{align*} \dd{y}{x}=f(x)=-\frac{M(x,y)}{N(x,y)}\\ \Leftrightarrow\\ M(x,y)dx+N(x,y)dy=0 \end{align*} $$