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

\vspace{.2in}

The general form of a second-order ode is

$$
y''(t)=f(t,y(t),y'(t)) 
$$

Usually, the differential equation is accompanied by 
initial conditions or boundary conditions.

\vspace{.3in}

{\bf Classes of Second-Order ODE That can be Solved Analytically} 

\vspace{.2in}

It is said that a differential equation is solved exactly if
the answer can be expressed in the form of an integral.
It may be possible to solve the following classes of
second-order ode exactly:

\begin{itemize}
\item linear

$$
f(t,y(t),y'(t))=p(t)y'(t)+q(t)y(t)+f(t)
$$

\item reducible (class 1)

$$
f(t,y(t),y'(t))=F(t,y'(t))
$$

The second derivative depends on the independent variable t
and the first derivative only.

\item reducible (class 2)

$$
f(t,y(t),y'(t))=F(y(t),y'(t))
$$

\end{itemize}

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