Definitions

ODE:  Ordinary Differential Equation

An Eq with d/dt or '

y'[t]+y[t]==0

Dependent Independent Variable

y'[t]+y[t]==0
ind  t
dep y

Order:

highest derivative
y'''[t]+y[t]==0
3rd order

Solution:

It is a soln if you plug in and it works. 

y'[t]+y[t]==0
 y[t] = (Subscript[c, 1])(E^(-t))  is a one parameter familly of sols on (-∞,∞)

Solution vs Function:

Solution is a function plus an interval

Implicit Solution:

y'[t]+y[t]==0
 exp sol  y[t] = (Subscript[c, 1])(E^(-t))  is a sol on (-∞,∞)
 imp sol ln(y)=-t + (Subscript[c, 1])

System of ODE:

y1' = 3 y1 + 2 y2 + 6 y3 + t
y2' = 12 y1 + 4 y2 -7 y3 + Sin[t]
y3' = 8 y1 + 4 y2 -5 y3 + Cos[t]

1st order linear system of three ODEs
 y' = A.y + (({{t}, {Sin[t]}, {Cos[t]}}))  where A = (({{3, 2, 6}, {12, 4, -7}, {8, 4, -5}}))

Initial Conditions:

y' = A.y + (({{t}, {Sin[t]}, {Cos[t]}}))  where A = (({{3, 2, 6}, {12, 4, -7}, {8, 4, -5}}))
y(1.2) = (({{0.1}, {-5}, {3.14}}))

Usefull Commands

Mathematica can analytically solve all the equations we are going to learn how to solve!

DSolve[{
y''[x]+2 y'[x]+12 y[x]==x
},
y[x],x]

DSolve[{
y''[x]+2 y'[x]+12 y[x]==x,
y[1]==3.2,
y'[1]==1
},
y[x],x]

Mathematica can numerically solve (almost) any ODE with enough initial conditions we are going to learn how to solve!

{y1Sol,y2Sol}={y1,y2}/.NDSolve[{
y1''[x]+ y1'[x]+12 y1[x]+ y2[x]==0,
y2'[x]+12 y2[x]+ y1[x]==Cos[x],
y1[0]==1,
y1'[0]==2,
y2[0]==1
},
{y1,y2},{x,0, 12}][[1]]
Plot[{y1Sol[x],y2Sol[x]},{x,0,12}]

Linear Algebra

You need to be able to use a calculator.  We are going to demo on a TI89 titanium.  All TIs are similar.  As always you can use Mathematica to check things.

Linear Solve :

A=(({{1.2, 2.3, 3.4}, {4.2, 5.1, 6.1}, {7.2, 8.2, 9.1}}));
b={3.2,3.6,7.8};
x=LinearSolve[A,b]
(* Checking *)
A.x
Norm[A.x-b]

Eigenstuff:

A=(({{1.2, 2.3, 3.4}, {4.2, 5.1, 6.1}, {7.2, 8.2, 9.1}}));
{λ,V}=Eigensystem[A];
MatrixForm[V]

Checking

i=1;
{λ[[i]] V[[i]], A.V[[i]]}
Norm[λ[[i]] V[[i]]- A.V[[i]]]

Table[
Norm[λ[[i]] V[[i]]- A.V[[i]]],
{i,1,3}]