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.

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

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

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

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

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

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.

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

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

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

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?

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?  

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) 