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

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