Solution of a First Order Ordinary Differential Equation

Here is an example of how to analytically solve our simple first order ODE by hand:

$\frac{d y (t)}{d t} = α y (t)$	Rate of change of y proportional to y itself
$\frac{d y (t)}{y (t)} = α d t \Rightarrow \int \frac{d y (t)}{y (t)} = \int α d t$	Rearrange terms and form integrals of both sides
$\ln (y) = A + α t \Rightarrow \ln (y_{0}) = A$	Assign integration constant and integrate both sides
$\ln (y) = \ln (y_{0}) + α t \Rightarrow \ln (\frac{y}{y_{0}}) = α t$	Use logarithm rules of addition
$y (t) = y_{0} e^{α t}$	Take exponential of both sides

Here is the general solution of a first order ODE:

$y' + p (x) y = g (x)$	Rearrange your equation into the form on the left
$y = \frac{1}{μ (x)} [\int_{}^{\infty} μ (s) g (s) d s + c]$	This is the solution in terms of the integrating factor
$μ (x) = e^{[\int_{}^{\infty} p (t) d t]}$	This is the integrating factor

Now we will look at several ODEs which have physical significance and we will discuss methods for solving them analytically using Maple.

CS_310