Definitions and Thms 8.1

General Solution: non-homogeneous Equation

If {X1, X2, X3,…,Xn} is a FS for
	({{X'[t], =, A[t].X[t]}})
and XP[t] solves
	({{X'[t], =, A[t].X[t], +, F[t]}})
then
	({{X[t], =XP[t]+C1 X1[t]}})+…+CN Xn[t] 
gives all solns of ({{X'[t], =, A[t].X[t], +, F[t]}}).

Reminders

Fundamental Set (FS)

A LI set of n solns of 
	({{X'[t], =, A[t].X[t]}})
is a fundamental set of solutions

Linear Dependence/Independence  (LD/LI)

A bunch of soln vectors {(Subscript[X, 1])(t),(Subscript[X, 2])(t),…,(Subscript[X, k])(t)}are LD if 
	(∑_(i=1)^k)(Subscript[C, i])(Subscript[X, i])=0
for(non-trivial constants)(Subscript[C, i]).  They are LI if they are not LD.

Wronskians (last bit of 8.1)

Linear Independence in Linear Algebra (Reminder)

The set of n n-vectors {(Subscript[X, 1]),(Subscript[X, 2]),…,(Subscript[X, n])} is LI if the n×n matrix 
	𝕏=(({{(Subscript[X, 1]), |, (Subscript[X, 2]), |, …, |, (Subscript[X, n])}}))
is non-singular.  A simple test for this is  
	Det[𝕏]≠0.
You can always solve a linear system 𝕏.C=x[0] if 𝕏 is non-singular.

Linear Dependence/Independence  in ODE

A bunch of soln vectors {(Subscript[X, 1])(t),(Subscript[X, 2])(t),…,(Subscript[X, k])(t)} are LI (on an interval) if the Wronskian
	W[t]=Det[(({{(Subscript[X, 1]), |, (Subscript[X, 2]), |, …, |, (Subscript[X, n])}}))]
is non-zero on the interval.

8.2 Const-Coeff Homog Systems: 
X'[t]=A.X[t]

Soln of y'[t]=a y[t]

y[t]=(Subscript[C, 1])(e^(a t))

Soln of X'[t]=A.X[t]

Try X[t]=V (E^(λ t))
X'[t]=λ V (E^(λ t))

λ V (E^(λ t)) = A.V (E^(λ t))
λ V  = A.V

What linear algebra problem is this?

Eigen stuff

Linear Algebra Reminder

Eigenvalues and Eigenvectors: A  is n×n matrix.

A vector V and scalar λ are an eigenvalue/eigenvector pair if
	A.V=λ V
Almost all n×n matrices have n LI eigenvectors.  
The eigenvalues/eigenvectors can be complex!
Matrices without enough eigenvalues are called Defective.

Computing Eigenvalues and Eigenvectors: A  is n×n matrix.

Type the matrix into your calculator or computer and hit the appropriate command.

Gen Sol of X'[t]=A.X[t]

Compute the n eigenvalues/eigenvectors of A {{(Subscript[λ, 1]),(Subscript[V, 1])},{(Subscript[λ, 2]),(Subscript[V, 2])},…,{(Subscript[λ, n]),(Subscript[V, n ])}}

Write down the set of solutions {(Subscript[V, 1])(e^((Subscript[λ, 1])t)),(Subscript[V, 2])(e^((Subscript[λ, 2])t)),…,(Subscript[V, n])(e^((Subscript[λ, n])t))}

Write down the general solution (Subscript[c, 1 ])(Subscript[V, 1])(e^((Subscript[λ, 1])t))+(Subscript[c, 2 ])(Subscript[V, 2])(e^((Subscript[λ, 2])t))+…+(Subscript[c, n ])(Subscript[V, n])(e^((Subscript[λ, n])t))

Sol of X'[t]=A.X[t] with X[0]=(Subscript[x, 0])

Write down the general solution X[t]=(Subscript[c, 1 ])(Subscript[V, 1])(e^((Subscript[λ, 1])t))+(Subscript[c, 2 ])(Subscript[V, 2])(e^((Subscript[λ, 2])t))+…+(Subscript[c, n ])(Subscript[V, n])(e^((Subscript[λ, n])t))

Plug in the initial condition to get  (Subscript[c, 1 ])(Subscript[V, 1])(e^((Subscript[λ, 1])0))+(Subscript[c, 2 ])(Subscript[V, 2])(e^((Subscript[λ, 2])0))+…+(Subscript[c, n ])(Subscript[V, n])(e^((Subscript[λ, n])0))=(Subscript[X, 0])

Solve the linear system 𝕍.C=(Subscript[X, 0]) for C=(({{(Subscript[c, 1])}, {(Subscript[c, 2])}, {⋮}, {(Subscript[c, n])}})) where 𝕍=(({{(Subscript[V, 1]), |, (Subscript[V, 2]), |, …, |, (Subscript[v, n])}}))

Write down the solution with the constants.

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

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