Introduction

Question 1

General Form of a First Order ODE

Answer 1

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)

Question 2

Nth Order ODEs

Answer 2

-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))

Question 3

Linear / Linear Homogeneous

First Order Definition

Answer 3

-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

Question 4

Linear Non-Homogeneous

First Order Definition

Answer 4

-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)

Question 5

Linear / Linear Homogeneous

Nth Order Definition

Answer 5

-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

Question 6

Linear Non-Homogeneous

Nth Order Definition

Answer 6

-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)

Question 7

Non-Linear

Definition

Answer 7

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

Question 8

Linear IAOI

Answer 8

-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

Question 9

Autonomous

Definition

Answer 9

-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))

Question 10

Non-Autonomous

Definition

Answer 10

-equations of the form:
x’n = f(t,x,x’, … , x’(n-1))
-for which f explicitly depends on t, are non-autonomous

Question 11

Solving First Order Autonomous ODEs

Answer 11

-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

Question 12

Can nth order linear autonomous equations be solved?

Answer 12

• 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

Question 13

Standard Form of a First Order System of ODEs

Answer 13

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

Question 14

Vector Form of a First Order System of ODEs

Answer 14

|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)

Question 15

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

Answer 15

-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))

Question 16

Linear / Linear Homogeneous

System of Equations Definition

Answer 16

-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

Question 17

Linear Non-Homogeneous

System of Equations Definition

Answer 17

-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

Question 18

Non-Linear

System of Equations Definition

Answer 18

-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

Question 19

Autonomous

System of Equations Definition

Answer 19

-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

Question 20

Non-Autonomous

System of Equations Definition

Answer 20

-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

Question 21

Converting a Non-Autonomous System to an Autonomous System

Answer 21

-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

Question 22

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

Answer 22

• 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

Question 23

What follows from the IVP and uniqueness theorem?

Answer 23

i) for any point in the extended phase space there is an integral curve passing through it (and thus for any point in phase space there is a trajectory passing through it
ii) the integral curves never intersect (uniqueness for solutions of the IVP)
iii) the trajectories (in the phase space) do not intersect each other (they may appear to intersect at fixed point of the system only)