Class 08/30/2021

Drag/Terminal Velocity

m \frac{d v}{d t} = - m g - k v (t), v (0) = v_{0} v (t) = \frac{- m g}{k} + (v_{0} + \frac{m g}{k}) e^{- k t / m}

Whether the object is initially moving up or down, it eventually begins to fall and approach terminal velocity. The instant before it strikes the ground it has the impact velocity.

Once the impact occurs, the initial value problem is no longer applicable.

First Order Linear Differential Equations

\frac{d y}{d x} + p (x) y = q (x), a < x < b

Homogeneous if and only if $q (x) = 0$ . If it cannot be written in this form then its nonlinear.

Example 1:

Determine the linearity and homogeneity of each of the following

$\frac{d y}{d x} = x y^{2}$ - nonlinear because of the quadratic $y^{2}$
$\frac{d y}{d x} = x^{2} y$ - linear with $p (x) = - x^{2}$ . It is also homogenous.
$cos (x) \frac{d y}{d x} + e^{x} y = tan (x)$
1. Linearize: $\frac{d y}{d x} + \frac{e ^{x}}{c o s ( x )} y = \frac{t a n ( x )}{c o s ( x )}$
2. Nonhomogeneous linear
$\frac{1}{y} \frac{d y}{d x} + x^{3} = sin (x)$
1. Can be linearized and is linear