Flashcards in Introduction Deck (23)

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1

## General Form of a First Order ODE

###
F(t, x, x') = 0

-where F is a continuous function of its arguments

-in particular F might not depend on t or x but must depend on x'

-usually we will assume that first order ODE can be solved wrt x' and use the following form:

x' = f(t,x)

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## Nth Order ODEs

###
-equations of the form:

F(t,x,x',x'', ... , x'n) = 0

-similarly to first order ODEs, we shall assume that nth order equations are resolved with respect to the highest derivative:

x'n = f(t,x,x', ... ,x'(n-1))

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##
Linear / Linear Homogeneous

First Order Definition

###
-the first order differential equation x'=f(t,x) is linear or linear homogeneous if function f(t,x) is a linear function of the dependent variable x

-i.e. if:

f(t,x) = p(t)x

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##
Linear Non-Homogeneous

First Order Definition

###
-the first order differential equation x' = f(t,x) is linear non-homogeneous if f(t,x) is of the form:

f(t,x) = p(t)x + q(t)

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##
Linear / Linear Homogeneous

Nth Order Definition

###
-we say that the equation

x'n = f(t,x,x', ... ,x'(n-1)) is linear homogeneous if function f is a linear function of variables x, x', x'', ... ,x'(n-1)

-i.e. if:

f(t,x,x', ... ,x'(n-1)) = Σpk(T)*x'k

-where the sum is taken between k=0 and k=n-1

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##
Linear Non-Homogeneous

Nth Order Definition

###
-we say that the equation

x'n = f(t,x,x', ... ,x'(n-1)) is linear non-homogeneous if function f is of the form:

f(t,x,x',...,x'(n-1))

= Σpk(T)*x'k + q(t)

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##
Non-Linear

Definition

### -equations which are not linear or linear non-homogeneous are called non-linear equations

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## Linear IAOI

###
-the equation:

x'n = f(t,x,x', ... , x'(n-1))

-is linear IAOI for any two solutions x1=x1(t) and x2 = x2(t), the linear combination:

x = A*x1 + B*x2

-is also a solution of the equation

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##
Autonomous

Definition

###
-the equation:

x'n = f(t,x,x', ... , x'(n-1))

-is called autonomous if function f does not depend on the independent variable t explicitly

-i.e. :

f = f(x,x', ... ,x'(n-1))

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##
Non-Autonomous

Definition

###
-equations of the form:

x'n = f(t,x,x', ... , x'(n-1))

-for which f explicitly depends on t, are non-autonomous

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## Solving First Order Autonomous ODEs

###
-first order autonomous ODEs are of the form:

x' = f(x)

-they are separable and can be solved:

dx/dt = f(x)

dx = f(x)dt

∫ 1/f(x) dx = t + C

-where C is an arbitrary constant of integration

12

## Can nth order linear autonomous equations be solved?

###
-in the case of linear (homogeneous or non-homogeneous) equations, nth order autonomous equations are equations with constant coefficients

-they can be solved explicitly using linear algebra

13

## Standard Form of a First Order System of ODEs

###
xk' = fk (t,x1,x2, ... , xn)

-for k = 1, ... , n

-here each fk is a specified function of dependent variables x1, .... , xn and independent variable t

14

## Vector Form of a First Order System of ODEs

###
|x' = |f (t, |x)

-where |x' is a column vector with entries x1', x2', ... , xn'

-and |x is the column vector x1, x2, .... , xn

-and |f is the column vector f1(t,|x) , f2(t,|x) , ... , fn(t,|x)

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## Writing Higher Order Equations (or systems of equations) in Terms of First Order

###
-any high order equation or a system of equations is equivalent to a first order equation

-take the equation:

x'n = f(t,x,x', ... , x'(n-1))

-if we denote x1=x , x2=x', x3=x''=x2', ... , xn=x'(n-1)

-then in new variables, the system can be written as |x'=|f(t,|x) where |f is a column vector with entries: x2, x3, ... , xn , f(t,x,x', ... , x'(n-1))

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##
Linear / Linear Homogeneous

System of Equations Definition

###
-a system of equations |x' = |f(t, |x) is said to be linear if vector function |f(t,|x) is a linear function of vector argument |x

-i.e. if:

|x' = f(t,|x) = A(t) |x

-where A(t) is an nxn matrix whose entries may depend on t

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##
Linear Non-Homogeneous

System of Equations Definition

###
-a system of equations |x' = |f(t, |x) is said to be linear non-homogeneous if vector function |f(t,|x) is of the form:

|x' = |f (t, |x) = |A(t) + |b(t)

-where |A(t) is an nxn matrix with entries that depend on t and |b(t) is a vector function of t with n components

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##
Non-Linear

System of Equations Definition

###
-a system of first order differential equations which cannot be written in the form:

|x' = f(t,|x) = A(t) |x

OR

|x' = |f (t, |x) = |A(t) + |b(t)

-is said to be non-linear

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##
Autonomous

System of Equations Definition

### -if in a system |x' = f(t,|x) the vector function |f is explicitly independent of t, then the system is said to be autonomous

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##
Non-Autonomous

System of Equations Definition

### -if in a system |x' = f(t,|x) the vector function |f is explicitly dependent on t, then the system is said to be non-autonomous

21

## Converting a Non-Autonomous System to an Autonomous System

###
-any non-autonomous system can be made autonomous in a bigger space

-introduce a new dependent variable xo which satisfies the equation xo'=1

-assuming initial condition xo(0)=0, the equation xo'=1 has a unique solution xo=t, but for now we will treat xo as just a dependent variable

-the non-autonomous system |x' = f(t,|x) is equivalent to autonomous system:

|y' = |F(|y)

-where |y is a column vector with entries: xo, x1, x2, ... , xn

-and |y' is a column vector with entries xo', x1', ..., xn'

-and |F is a column vector with entries: 1, f1(xo,|x), f2(xo,|x), ... , fn(xo,|x)

-a general solution of this autonomous system of n+1 equation depends on n arbitrary constants since constant xo(0) is fixed as 0

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## Is converting a non-autonomous system to an autonomous system useful?

###
-it is not always useful to convert a non-autonomous system to autonomous

-for example, a linear non-autonomous system may become non-linear after the conversion

-usually a linear system is easier to study than a nonlinear one, even if it is non-autonomous

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