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6.3Separation of Variables and the Logistic Equation
Objective:
Solve differential equations that can be solved by separation of variables.
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Separation of VariablesAn equation of the form
is said to be separable and can be solved using separation of variables.
Original DE Rewrite
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Example 1: Find the general solution of
𝑥2+3 𝑦𝑑𝑦𝑑𝑥
=0
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Example 2: Find the general solution of
sin 𝑥 𝑦 ′=cos 𝑥
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Example 3: Find the general solution of
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Homogeneous Functions
is a homogenous function of degree if
(1)
(2)
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Homogeneous Differential Equations
A homogeneous differential equation is an equation of the form
where and are homogeneous functions of the same degree.
(1)
(2)
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Change of Variables for Homogenous Equations
If is homogenous, then it can be transformed into a DE whose variables are separable by the substitution
where is a differentiable function of .
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Example 1
(𝑥2− 𝑦2 )𝑑𝑥+3 𝑥𝑦 𝑑𝑦=0
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Example 2
𝑦 ′=𝑥3+𝑦3
𝑥 𝑦❑2
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Example 3
𝑦 ′=𝑥2+𝑦 2
2 𝑥𝑦
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Example 4
𝑦 ′=3𝑥+2 𝑦
𝑥
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Orthogonal Trajectories
Common problem in electrostatics, thermodynamics, and hydrodynamics
Involves finding a family of curves, each of which is orthogonal to all members of a given family of curves.
Ex. and Two such families are said to be mutually
orthogonal, and each curve in one family is called an orthogonal trajectory of the other family.
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Example 5
Describe the orthogonal trajectories for the family of curves given by
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Logistic Differential Equation
Exponential growth is unlimited, but when describing a population, there often exists some upper limit L past which growth cannot occur. This upper limit L is called the carrying capacity, which is the maximum population y(t) that can be sustained or supported as time t increases.
A model that is often used to describe this type of growth is the logistic differential equation
where k and L are positive constants.
From the equation, you can see that if y is between 0 and the carrying capacity L, then dy/dt > 0, and the population increases.
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Logistic Differential Equation
If y is greater than L, then dy/dt < 0, and the population decreases. The graph of the function y is called the logistic curve.
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Example 6 – Deriving the General Solution
Solve the logistic differential equation
Solution: