Section 1.2  Solutions and Initial Value Problems

objectives

• to be able to determine whether given functions are solutions to differential equations
• to be able to determine whether given functions are solutions to initial value problems
• to be able to determine whether Theorem 1 implies a given initial value problem has a unique solution

• nth order differential equation := 

 
• explicit solution (defined on some interval)

 

 
 
 

• implicit solution

 

 
 
 

• Implicit Function Theorem

 

 
 
 

• initial value problem

 

 
 
 

• Theorem 1: existence and uniqueness of solution (first order, initial value)

 

 
 
 
 
 

determining if a function is an implicit solution to a differential equation, #9,11
 
 
 
 
 
 
 
 
 
 
 

one parameter family of solutions, #17
 
 
 
 
 
 

finding solutions of the form f(x) = x^m, #21.a
 
 
 
 
 
 
 

using Theorem 1 to determine if there is a unique solution, #25,27
 
 
 
 
 
 
 

some problems have no solutions, #19
 
 
 
 
 
 

initial value problem with more than one solution, #29
 
 
 
 
 
 
 
 
 
 

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QUESTION OF THE DAY

Use the technique of partial fractions to evaluate .

SOLUTION

 

SOLUTION.  Rewrite the rational fraction as a sum of rational fractions.