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Differential Equations
Sec 6.3: Separation of Variables
Separation of Variables
all x terms can be collected with dx, and all y terms with dy, and a solution can be obtained by integration.
Such equations are said to be separable, and the solution procedure is called separation of variables.
Examples: Separation of Variables
2 3 0dy
x ydx
'
21y
xy
e
sin ' cosx y x
Practice Problem
Find the general solution of the differential equation:
2 4dy
x xydx
Practice Problem: Particular Solution Given the initial condition y(0) = 1, find the
particular solution of the differential equation:
2 2 1 0xxy dx e y dy
Practice Problem:Particular Solution Curve Find the equation of the curve that passes
through the point (1, 3) and has a slope of y/x2 at any point (x, y).
Application: Wildlife Population
The rate of change of the number of coyotes N(t) in a population is directly proportional to 650 – N(t), where t is the time in years. The population was initially at 300. After 2 years, the population increased to 500. Find the population when t = 3.
Luxury is a necessity that begins when necessity ends.
--- Coco Chanel
Fin…