Definitions and Thms 8.1

Linear System (1st order) non-homogeneous

({{x1'[t], =, t x1[t], +, 3 x2[t], +, x3[t], +, Sin[t]}, {x2'[t], =, 4x1[t], -, (t^2) x2[t], +, 3 x3[t], -, 4t}, {x3'[t], =, 2x1[t], -, Cos[t] x2[t], +, 1 x3[t], +, (t^3)}})

Matrix Version

	({{X'[t], =, A[t].X[t], +, F[t]}})
	
where 
	X[t]=(({{x1[t]}, {x2[t]}, {x3[t]}})), F[t]=(({{Sin[t]}, {-4 t}, {(t^3)}})), and ({{A, =, (({{t, 3, 1}, {4, -(t^2), 3}, {2, -Cos[t], 1}}))}})

Existence and Uniqueness Thm

({{X'[t], =, A[t].X[t], +, F[t]}})  and ({{x[(Subscript[t, 0])], =, (Subscript[x, 0])}})
Has a unique soln on provided A and F are continuous on an interval containing (Subscript[t, 0]).

Constant coefficient Linear System (1st order) non-homegeneous

({{x1'[t], =, 5x1[t], +, 3 x2[t], +, x3[t], +, Sin[t]}, {x2'[t], =, 4x1[t], -, 4 x2[t], +, 3 x3[t], -, 4t}, {x3'[t], =, 2x1[t], +, 2 x2[t], +, 1 x3[t], +, (t^3)}})

Matrix Version

	({{X'[t], =, A.X[t], +, F[t]}})
	
where 
	X[t]=(({{x1[t]}, {x2[t]}, {x3[t]}})), F[t]=(({{Sin[t]}, {-4 t}, {(t^3)}})), and ({{A, =, (({{5, 3, 1}, {4, -4, 3}, {2, 2, 1}}))}})

Constant coefficient Linear System (1st order) homegeneous

({{x1'[t], =, 5x1[t], +, 3 x2[t], +, x3[t]}, {x2'[t], =, 4x1[t], -, 4 x2[t], +, 3 x3[t]}, {x3'[t], =, 2x1[t], +, 2 x2[t], +, 1 x3[t]}})

Matrix Version

	({{X'[t], =, A[t].X[t]}})
	
where 
	X[t]=(({{x1[t]}, {x2[t]}, {x3[t]}})), and ({{A, =, (({{5, 3, 1}, {4, -4, 3}, {2, 2, 1}}))}})

Superposition: Adding up solns

If {X1, X2, X3,…,Xk} are a bunch of solution vectors for ({{X'[t], =, A[t].X[t]}})  note HOMOGENEOUS then
C1 X1 +C2 X2 + C3 X3+…+Ck Xk
is another solution

Linear Dependence

A bunch of solutions {(Subscript[X, 1]), (Subscript[X, 2]), (Subscript[X, 3]),…,(Subscript[X, k])} for ({{X'[t], =, A[t].X[t]}}) are linearly dependent if
	(Subscript[C, 1]) (Subscript[X, 1])+(Subscript[C, 2]) (Subscript[X, 2]) +…+(Subscript[C, k]) (Subscript[X, k]) = 0
with some non-zero Cs. 
They are linearly independent if they are not linearly dependent.

Fundamental Set

A linearly independent set of n solutions of a system of n homogeneous 1st order ODEs
({{X'[t], =, A[t].X[t]}})
is a fundamental set of solutions

General Solution: Homogeneous Equation

If {X1, X2, X3,…,Xn} is a fundamental set for a system of n homogeneous 1st order ODEs
({{X'[t], =, A[t].X[t]}})
then any solution is of the form
	({{X[t], =, C1 X1[t], +C2 X2[t]}})+…+CN Xn[t]
This is called a general solution.

Particular Solution

Any solution (without constants) for a system of n homogeneous 1st order ODEs
	({{X'[t], =, A[t].X[t]}})+F[t]
is called a particular solution.

General Solution: non-homogeneous Equation

If {X1, X2, X3,…,Xn} is a fundamental set for a system of n homogeneous 1st order ODEs
({{X'[t], =, A[t].X[t]}})
and XP[t] is a solution of
({{X'[t], =, A[t].X[t], +, F[t]}})
then any solution of ({{X'[t], =, A[t].X[t], +, F[t]}}) is of the form
	({{X[t], =XP[t]+C1 X1[t]}})+…+CN Xn[t]

Usefull Commands

Mathematica can analytically solve all the equations we are going to learn how to solve! Lets look at a baby example.

{y1[t],y2[t]}/.DSolve[{
y1'[t]== y1[t]+4y2[t],
y2'[t]== y1[t]- 2y2[t],
y1[0]==1,y2[0]==2},
{y1[t],y2[t]},t][[1]]//Expand

Linear Algebra

We solve 1st order constant coefficient linear systems using eigenvalues and eigenvectors.

Eigenvalues: Av=λv

A=(({{1, 4}, {1, -2}}));
Eigensystem[A]