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{\Large\bf Analytic Methods for Solving First-Order ODE}

\vspace{.2in}

The general form of a differential equation is

$$
y'(t)=f(t,y(t)) \hspace{.2in} y(t_0)=y_0
$$

Here t is the independent variable and y is the dependent variable.
The goal is to find the unknown function y(t). 

\vspace{.3in}

{\bf Classes of First-Order ODE that can be Solved Analytically} 

It is said that a differential equation can be solved exactly if
the answer can be expressed in the form of an integral.
The following classes of first-order ode can be solved exactly: 

\begin{itemize}
\item directly integrable

$$
f(t,y(t))=g(t) 
$$

(derivative function depends on independent variable only)

\item linear

$$
f(t,y(t))=g(t)y(t)+h(t)
$$

\item separable

$$
f(t,y(t))=g(t)h(y(t))
$$

\item homogeneous

$$
f(t,y(t))=g(y(t)/t)
$$

\item exact

$$
f(t,y)dt+g(t,y)dy=0,
$$

where f_y=g_t

\item Bernoulli

$$
f(t,y(t))=g(t)y(t)+h(t)[y(t)]^n
$$

where n doesn't equal 1.

\end{itemize}

This list is not exhaustive, but covers core classes of
first-order ode that can be solved exactly.

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