# Boundary Value Problems/Review: ODEs

For more of an introduction to ODEs you are referred to Boundary Value Problems/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 $\infty . Initial value: $x(0)=-2$ 1. Differential eq: ${\frac {dx}{dt}}=5x+2$ where $x(t)$ for all $\infty . Initial value: $x(3)=-2$ 