Math 150B Lecture Notes by Professor David Nusbaum Prerequisite material:You are expected to know all the material in the review packet. It is recommended that you study the instructor’s online flash cards at Cram.com until you get everything100% correct. Then review the flash cards at least once per week throughout the semester. Section 6.2 Differential Equations; Growth and Decay: Def:A differential equation in xand yis an equation that involves x,y, and ____________________________ Def:A function _________________ is called a solutionof a differential equation if the equation is satisfied when yand its derivatives are replaced by _________________________________________________. For example, 2xy3e−=is a solution of the differential equation y2y0+=. It can be shown that every solution of this differential equation is of the form _______________ This is called the _______________________________. Def:The ___________ of a differential equation is the highest order derivative in the equation. y2y0+=is a __________ order differential equation. s (t)32= −is a ______________ order differential equation. A differential equation that models a constant rate of change isyk =, where kis constant. For Examples 1 to 6, solve the differential equation Example 1: yk =

Page 2

=

Page 3 Example 6a: 2(1x )y2xy0+−=Example 6b: Find the specific solution of the equation above that is satisfied by the point (2, 15). Example 7: (You try) Find the function yf(t)=with the given first derivative whose graph contains the point (0, 10). dy9tdt= −

Page 4 Example 8: The rate of change of P is proportional to P. Whent0=, P5000=and when t1=, P4750=. the value of P when t5=In many applications, the rate of change of a variable yis proportional to the value of y. This is written as dykydt=. This is an exponential model, and the general solution of this equation is __________________ _____ is the initial value of yand _____ is the proportionality constant. Exponential growth occurs if ___________ Exponential decay occurs if _____________ Money that earns continuously compounded interest follows an exponential growth model. Example 9: An initial investment of $2500 earns 4% interest, compounded continuously. (a) Find the amount after 10 years. (b) Find the time it takes to double in value. Find .

Page 5