Introduction Flashcards
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)
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))
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
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)
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
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)
Non-Linear
Definition
-equations which are not linear or linear non-homogeneous are called non-linear equations
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
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))
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
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
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
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
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)
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))
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
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
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
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
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
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
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
What follows from the IVP and uniqueness theorem?
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)
