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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Linear and Bernoulli Equations">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Tue Apr 01 16:43:43 1997}
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\begin{document}


\section{Linear and Bernoulli Equations\label{Linear and Bernoulli Eqautions}
}

\begin{definition}
\vspace{1pt}A first-order differential equation is called \textbf{linear 
\label{Linear Differential Equation}}if it can be written in the form 
\[
\frac{dy}{dx}+P(x)y=Q(x)\text{.}
\]
\end{definition}

This equation may be solved by multiplying both sides by the function $%
e^{\int Pdx}$, which is called an \textbf{integrating factor. \ }The left
side becomes the derivative of $ye^{\int Pdx}$. Integration of both sides
will yield the solution.

\begin{theorem}
The general solution of a first order linear equation is given by 
\[
y=e^{-\int Pdx}\left[ \int e^{\int Pdx}Qdx+c\right] 
\]
where $c$ is an arbitrary constant.
\end{theorem}

\begin{definition}
A \textbf{Bernoulli Differential Equation\label%
{Bernoulli Differential Equation}} is a differential equation of the form 
\[
\frac{dy}{dx}+P(x)y=Q(x)y^{n}
\]
where $n$ is an integer.
\end{definition}

For $n=1$, this equation reduces to a separable linear differential equation.

\begin{theorem}
Suppose $n\neq 1$. Then the transformation $u=y^{1-n}$ reduces the equation
to a first-order linear differential equation in $u$.
\end{theorem}

\begin{example}
The equation $y^{\prime }+2xy=3x$ is a first order linear differential
equation with $P(x)=2x$, $Q(x)=3x$. The integrating factor for this equation
is $e^{\int 2xdx}=\allowbreak e^{x^{2}}$. Write 
\[
\allowbreak e^{x^{2}}\frac{dy}{dx}+\allowbreak e^{x^{2}}2xy=3\allowbreak
xe^{x^{2}}
\]
Integrating both sides yields 
\[
\int \left( \allowbreak e^{x^{2}}\frac{dy}{dx}+\allowbreak
e^{x^{2}}2xy\right) dx=\allowbreak e^{x^{2}}y=\int 3\allowbreak
xe^{x^{2}}dx=\allowbreak \frac{3}{2}e^{x^{2}}+c
\]
so that 
\[
e^{x^{2}}y=\frac{3}{2}e^{x^{2}}+c
\]
and the solution is 
\[
y=\frac{3}{2}+ce^{-x^{2}}
\]
\end{example}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Exact solutions}{}{}{../../Help/DM10-1.tex#Differential
equation}}

{\small \hyperref{Solve ODE, exact}{}{}{../../Help/DM10-1.tex#Solve ODE}}

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\noindent \hrulefill

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