# Category: First-Order ODE

## Separable Differential Equations

Consider the differential equation $$${y}'={f{{\left({t},{y}\right)}}}$$$, or $$$\frac{{{d}{y}}}{{{d}{t}}}={f{{\left({t},{y}\right)}}}$$$.

If the function $$${f{{\left({t},{y}\right)}}}$$$ can be written as the product of the function $$${g{{\left({t}\right)}}}$$$ (function that depends only on $$${t}$$$) and the function $$${u}{\left({y}\right)}$$$ (function that depends only on $$${y}$$$), such a differential equation is called separable.

## Homogeneous Equations

If in the differential equation $$${y}'={f{{\left({t},{y}\right)}}}$$$, the function $$${f{{\left({t},{y}\right)}}}$$$ has the property that $$${f{{\left({a}{t},{a}{y}\right)}}}={f{{\left({t},{y}\right)}}}$$$, such a differential equation is called homogeneous.

## Exact Equations

The differential equation $$${M}{\left({x},{y}\right)}{d}{x}+{N}{\left({x},{y}\right)}{d}{y}={0}$$$ is exact, if there exists a function $$${f{}}$$$ such that $$${d}{f{=}}{M}{\left({x},{y}\right)}{d}{x}+{N}{\left({x},{y}\right)}{d}{y}$$$.

## Linear Differential Equations

A first-order linear differential equation has the form $$${y}'+{p}{\left({t}\right)}{y}={q}{\left({t}\right)}$$$.

To solve it, rewrite it in the differential form: $$$\frac{{{d}{y}}}{{{d}{t}}}+{p}{\left({t}\right)}{y}={q}{\left({t}\right)}$$$, or $$${\left({p}{\left({t}\right)}{y}-{q}{\left({t}\right)}\right)}{d}{t}+{d}{y}={0}$$$.

## Bernoulli Equations

A Bernoulli equation has the form $$${y}'+{p}{\left({t}\right)}{y}={q}{\left({t}\right)}{{y}}^{{n}}$$$ where $$${n}$$$ is a real number.

Using the substituion $$${z}={{y}}^{{{1}-{n}}}$$$, this equation can be transformed into a linear one.