Chapter 2 - First Order Differential Equation
Section 2.
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Section 2.1 - Autonomous ODE and Critical Point
Section 2.1
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Definition (Autonomous ODE)
An ODE in which the independent variable does not appear
explicitly is called autonomous ODE.
dy
= f (y)
dx
Section 2.1
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Definition (Autonomous ODE)
An ODE in which the independent variable does not appear
explicitly is called autonomous ODE.
dy
= f (y)
dx
Examples: (a)
Section 2.1
dy
= 1 + y3
dx
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(b)
dy
x
=−
dx
y
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Definition (Critical Point)
If f (c) = 0 for a real number c in the autonomous ODE
dy
= f (y), then c is called the critical point of the ODE
dx
(same as equilibrium point or stationary point).
Section 2.1
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Definition (Critical Point)
If f (c) = 0 for a real number c in the autonomous ODE
dy
= f (y), then c is called the critical point of the ODE
dx
(same as equilibrium point or stationary point).
Remark
dy
= f (y), then y(x) = c is a constant
If c is a critical point of
dx
solution of the autonomous ODE. This type of solution is
known as equilibrium solution.
Section 2.1
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Section 2.2 - Separable ODE
Section 2.2
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The general form of the 1st order ODE looks like
Section 2.2
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dy
= f (x, y).
dx
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The general form of the 1st order ODE looks like
dy
= f (x, y).
dx
Definition (Separable ODE)
dy
If f (x, y) = g(x)h(y) or
= g(x)h(x), then this ODE is called
dx
separable.
Section 2.2
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The general form of the 1st order ODE looks like
dy
= f (x, y).
dx
Definition (Separable ODE)
dy
If f (x, y) = g(x)h(y) or
= g(x)h(x), then this ODE is called
dx
separable.
dy
dy
Examples: (a)
= ex+y
(b)
= sin (x + y)
dx
dx
Section 2.2
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The general form of the 1st order ODE looks like
dy
= f (x, y).
dx
Definition (Separable ODE)
dy
If f (x, y) = g(x)h(y) or
= g(x)h(x), then this ODE is called
dx
separable.
dy
dy
Examples: (a)
= ex+y
(b)
= sin (x + y)
dx
dx
Remark
dy
dy
If
= f (y) or
= g(x)h(y) then y = c is also a solution of
dx
dx
the ODE if f (c) = 0 or h(c) = 0
Section 2.2
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Examples
Solve the following ODEs:
dy
= e3x+2y
(a)
dx
Section 2.2
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(b)
p
dy
= x 1 − y2
dx
Section 2.2
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(c)
dP
= P − P2
dt
Section 2.2
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Section 2.2
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(d)
xy + 3x − y − 3
dy
=
dx
xy − 2x + 4y − 8
Section 2.2
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Section 2.2
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(d) (ex + e−x )
Section 2.2
dy
= y2
dx
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Section 2.2
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