# Boundary Value Problems/Introduction

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An example of a boundary value problem in one dimension is the second order linear differential equation:

${\displaystyle ay''+by'+cy=0}$ with the end conditions of ${\displaystyle y(0)=0}$ and ${\displaystyle y(L)=0}$.

For a simple problem let a=1, b=0, c=1 and ${\displaystyle L=\pi }$. The resulting differential equation is ${\displaystyle y''+y=0}$ with boundary conditions ${\displaystyle y(0)=0}$ and ${\displaystyle y(\pi )=0}$. A solution is ${\displaystyle y(x)=sin({{\pi x} \over L})}$ .

A plot of this solution is shown below. Note that the solution satisfies the boundary conditions.

For more information about ordinary differential equations and methods for solving them, use the following link ODE