Boundary Value Problems/Review: ODEs

Return to Boundary Value Problems

For more of an introduction to ODEs you are referred to ../BVP-Ordinary-Differential-Equations and for examples please look at Examples of ordinary differential equations

First order differential equation:
Let $$ t \in [a,b] $$ the equation $$\frac{dy}{dt}=k(t)y+f(t) $$

is a linear ordinary (single independent variable) differential equation of first order (first derivative). If $$ f(t)=0 $$ on $$ [a,b] $$ the equation is called "homogeneous" differential equation. Otherwise it is called nonhomogeneous otherwise.

Examples of homogeneous and non-homogeneous:

$$ (Homogeneous) \frac{d}{dt} y = 10 y $$ where $$ k(t)=10 $$ is a constant coefficient.

$$ (Homogeneous) \frac{d}{dt} y = t y $$ where $$ k(t)= t $$ is a coefficient that increases as $$ t $$ increases.

$$ (Nonhomgeneous) \frac{d}{dt} y = 10 y + sin(t) $$ where $$f(t)=sin(t)$$.

Solving homogeneous ODE with constant coefficients: Separable case
Solving $$ \frac{d}{dt} y = 10 y $$

$$ \frac{1}{y} \frac{d}{dt} y = 10  $$

$$ \int \frac{1}{y} \frac{dy}{dt} dt= \int 10 dt $$ $$ ln(|y|) = 10t + C_1 $$ where $$C_1$$ is the integration constant.

$$ e^{ln(|y|)} = e^{10t + C_1} $$

$$ y(t) = e^{C_1}e^{10t} $$

$$ y(t) = C_2e^{10t} $$

After integration there is a constant, an unknown. This will happen each time you integrate. To determine this unknown for a particular application you will need another piece of information. Typically it is another equation that requires $$ y(0)= intial value $$. This condition with the above differential equation defines an Initial Value Problem. For this problem the value of C_2 is determined by using an intial condition such as $$ y(0)= 5 $$. Then $$ y(0)=C_2 e^{0}=C_2 = 5 $$ and making the substitution provides the solution $$ y(t) = 5 e^{10t} $$

Student work
Solve the following IVPs:


 * 1) Differential eq: $$  \frac{dx}{dt} = 5x $$ where $$x(t)$$ for all $$ \infty < t <\infty $$. Initial value: $$ x(0)=-2 $$

Click here for solution: ../BVP-solution-1


 * 1) Differential eq: $$  \frac{dx}{dt} = 5x + 2 $$ where $$x(t)$$ for all $$ \infty < t <\infty $$. Initial value: $$ x(3)=-2 $$

Click here for solution: ../BVP-solution-2

Pre-class work:
Complete each of the following:
 * Read the following:
 * Pages 1-10 of Powers
 * Lecture notes (web page)
 * (View) Associated Lecture Video
 * Work the following problems ../BVP-Exercise 1-1:Problems and Solutions