1. MTH 253 Calculus (Other Topics)
Chapter 9 – Mathematical Modeling with
Differential Equations
Section 9.1 – First-Order Equations
and Applications
Copyright © 2006 by Ron Wallace, all rights reserved.

2. Terminology … Differential Equation
An equation involving one or more derivatives of an unknown function.
Examples:

3. Terminology … Order of a Differential Equation
The highest order derivative in the equation.
Examples:
Order = 2
Order = 3

4. NOTE: # of constants in the general solution = order of the DifEq
Terminology …
Solution of a Differential Equation
A function whose derivatives make the differential equation a true statement.
A general solution is a function w/ parameters that represents ALL possible solutions.
Example:
A solution:
General solution:
NOTE: # of constants in the general solution = order of the DifEq

5. NOTE: # of initial conditions of an IVP = order of the DifEq
Terminology …
Initial-Value Problem
A differential equation with conditions that determine a unique solution.
Example:
General solution:
IVP solution:
NOTE: # of initial conditions of an IVP = order of the DifEq

6. A Differential Equations Course
The development and study of methods for solving differential equations and IVP’s.
Categorizing Differential Equations
Explicit methods
Find solutions and general solutions
Numerical methods
Find a set of ordered pairs that approximate the solution of an IVP.
Applications

7. 1st Order Separable Equations
The variables can algebraically be separated, giving the form …
Example: 
NOTE: Since we multiplied by x and divided by y2,
we must assume that x0 and y0.

8. Solving 1st Order Separable Equations
Algebraically separate the variables.
Integrate both sides of the equation.
Solve for y (if possible).
Example:     
Check the solution?

9. 1st Order Linear Equations
Equations that can be algebraically manipulated into the form …
Example: 

10. Solving 1st Order Linear Equations
Find
Solution is ...
Example: 

11. Applications: Mixing Problems
A tank contains a solution with a known amount of a substance (y).
A solution with a known concentration of the substance is entering the tank at a known rate.
The tank is draining at a known rate.
The solution in the tank is kept thoroughly stirred.
How much of the substance is in the tank at any point in time?

12. Applications: Mixing Problems
A polluted lake contains 1 lb of mercury salts per 100,000 gallons of water. The volume of the lake is 560,000 gallons. Water is pumped from the lake at 1000 gallons/hour and replaced by fresh water a the same rate. How much mercury salts is in the lake after 100 hours of pumping?  

13. Applications: Free Falling Object
An object is in free fall.
At time t = 0, the height is s0 and velocity is v0.
Two forces are acting on the object:
Gravity: FG = -mg (mass times gravity [32 ft/sec2])
Air Resistance: FR = -cv (v is the object’s velocity)
FR is called drag
c is a positive constant depending on the shape of the object
Newton’s Second Law of Motion (F = ma) 