Chapter 1 : Introduction

Section 1.1 Some Basic Mathematical Models; Direction Fields

What is a differential equation (DE)?

Basically, differential equations are equations involving derivatives.They can model dissipation of heat,wave propagation,fluid motion, flow of current in electric circuits, etc.

Examples of DEs:

1) Newton's 2nd Law : $F = m * a$ = $m * (dv/dt)$ Since $a = dv/dt$

2) $x*y'' - y' + (4*x^3)*y = 0$

3) $dy/dx = 2y$

NOTE: $dy/dx = y'$

How do we solve DEs?

There are many ways to solve DEs:

1) Using Direction Fields (Explained in another chapter)
2) Using Analytic Approach : Find exact form of solutions
3) Using Numerical Approach (Explained in another chapter)

For instance, solving the equation $F = m * (dv/dt)$ involves finding a function $v = v(t)$ that satisfies that equation.

Section 1.2 Solutions of Some Differential Equations

In this section, we will learn that the rules of calculus can help us solving DEs

Consider the equation : $dp/dt = 0.5p - 450 $

Let's try to find the solutions of this equation: To solve the equation, we need to find a function p(t) that, when substituted into the equation will reduce it into an obvious relationship.

We can rewrite it in the form : $dp/dt = (p - 900)/2 $

If p is not equal to 900: $(dp/dt)/(p-900) = 1/2 $

Using the chain rule: $d/dt (ln |p - 900|) = 1/2 $

Integrating both sides: $ ln |p - 900| = (1/2)* t + C $

Taking the exponential of both sides: $ |p - 900| = e^(t/2 + C) = (e^C)* (e^(t/2))$

Or: