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:

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

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

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

Solving homogeneous ODE with constant coefficients: Separable case

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

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

$\int \frac{1}{y}\frac{dy}{dt}dt=\int 10dt$

$ln(|y|)=10t+C_1$

where

$C_1$

is the integration constant.

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

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

$y(t)=C_2{e}^{10t}$

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 $y(0)=intialvalue$. 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 $y(t)=5e^{10t}$

Student work

Solve the following IVPs:

1. Differential eq: $\frac{dx}{dt}=5x$ where $x(t)$ for all $\mathrm{\infty }<t<\mathrm{\infty }$. 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 $\mathrm{\infty }<t<\mathrm{\infty }$. 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]]