Chapter 2 Solutions of 1 st Order Differential Equations.

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Presentation transcript:

Chapter 2 Solutions of 1 st Order Differential Equations

Sec 2.2 – Separable Equations  The simplest form of a 1 st order equation for which a solution is available  Begin with the usual setup  If H can be written in the form  Then the original equation can be written  And so  Now we can just integrate both sides, to get

Example  Suppose  Take a minute and see if you can guess the solution  What function when differentiated = 2x times itself?  To solve, separate the variables by diving both sides by y, multiplying by dx to get

Example – Section 2.2 #13  Suppose

Sometimes we can’t solve for y explicitly  Example:  Then  So  Which we can write  Usually simplified to  Do you recognize this graph?  Can’t solve for an explicit y; the given form is an implicit solution

Applications of Separable DE  From Calculus:  Natural Growth and Decay  Population growth (bacteria, world population)  Decay (radioactive decay, drug dissipation)  These conform to General Exponential Growth Equation:  Where k is the growth constant (pos or neg)  (Continuously) compounded interest  Cooling & Heating (Newton’s Cooling Law)  We will take these up in Ch 4

The equations (notation from the particular science)  Natural growth:  Natural Decay:  Drug dissipation:  Compound Interest:  Cooling & Heating: