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.