# Boundary Value Problems/Introduction to BVPs

## Objective

Introduce Boundary value problems for a single independent variable.

## Approach

• What is a Boundary Value problem?
• Solution of a Boundary Value Problem is directly related to solution of an Initial Value Problem. So let's review the material on IVPs first and then make the connection to BVPs.
• Details of solving a two point BVP.

## Initial Value Problems

For a single independent variable $x$  in an interval $I:a , an initial value problem consists of an ordinary differential equation including one or more derivatives of the dependent variable, $y$ ,

$y^{(n)}+p_{n-1}(x)y^{(n-1)}+...+p_{1}(x)y'(x)+p_{0}y(x)=f(x)$

and $n$  additional equations specifying conditions on the solution and the derivatives at a point $x_{0}\in I$

$y^{(n-1)}(x_{0})=b_{n-1}$ , ..., $y'(x_{0})=b_{1}$ , $y(x_{0})=b_{0}$

Example:

The differential equation is $y'=x$  (First order differential equation.) and the initial condition at $x=0$  is given as $y(0)=1$  .

Solution:

$\int y'dx=\int xdx$

$y={\frac {x^{2}}{2}}+C$ .

When,$x=0$  $1=C$  and $y={\frac {x^{2}}{2}}+1$

Get out a piece of paper and try to solve the following IVP in a manner similar to the preceding example:

$y'=xy$  and the initial condition at $x=0$  is given as $y(0)=3$  .

A second order ODE example:

The differential equation is $y''+5y'+4y=0$  (Second order differential equation.) and the two initial conditions at $x=0$  given as $y(0)=1,y'(0)=-2$  .

Solution:

Assume the solution has the form $y=e^{rx}$

$y'=re^{rx},y''=r^{2}e^{rx}$

$y''+5y'+4y=r^{2}e^{rx}+5re^{rx}+4e^{rx}$

$0=r^{2}e^{rx}+5re^{rx}+4e^{rx}$

$0=r^{2}+5r+4$  The characteristic polynomial. Solve for "r".

${\begin{array}{c}r_{1}=4\\r_{2}=1\end{array}}$

See the Wikipedia link for more on Initial Value Problems

## Two point BVPs for an ODE

Begin with second order DEs, $x''=f(t,x,x')$ , with conditions on the solution at $t=a$  and $t=b$ .

${\frac {d^{2}x}{dt^{2}}}+p(t){\frac {dx}{dt}}+q(t)x(t)=f(t)$  with $a_{0}x(a)+a_{1}x'(a)=g$  and $b_{0}x(b)+b_{1}x'(b)=h$  on the interval $I_{ab}=\{x|a\leq t\leq b\}$

## Example

${\frac {d^{2}x}{dt^{2}}}+4{\frac {dx}{dt}}+2x(t)=f(t)$  with $x(0)=0$  and $x(1)=0$  on the interval $I_{ab}=\{x|0\leq t\leq 1\}$

See the wikipedia topic