Definitions and Thms 1.2

Standard Form (1st order and second order) Solved for Highest Der

x y'[x] + y[x] == 3  (Not Std) y'[x] + 1/x y[x] == 3/x 1st order Std form and linear

y''[x] + x y'[x] + ((y[x])^2)= 3  2nd order Standard Form NOT linear 

Generic 1st order linear 
y'[x] + p[x] y[x] = f[x]

Generic 2nd order linear 
y''[x] + p[x] y'[x] + q[x] y[x] = f[x]

IVP:  Initial Value Problem (1st order and second order)

1st order 
y'[x] = f(x,y[x])  y[x0]=y0

2nd Order
y''[x] = f(x,y[x],y'[x])  y[x0]=y0 and y'[x0]=y1

Interval of Existence: Keep going from the x0 untill you hit a problem.

Thm: Existence and Uniqueness

y'[x] =f[x,y[x]] and y[(Subscript[x, 0])]=(Subscript[y, 0]) has one and only one solution is f is nice near {(Subscript[x, 0]),(Subscript[y, 0])}.

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]

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]+2 y1'[x]+12 y2[x]==x,
y2'[x]+ Sin[y1[x] y2[x]]==Cos[ArcTan[(x^2)]],
y1[0]==1,
y1'[0]==2,
y2[0]==1
},
{y1,y2},{x,0, 123}][[1]]
Plot[{y1Sol[x],y2Sol[x]},{x,0,123}]

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, 5.3, 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

Eigenstuff:

A=(({{1.2, 5.3, 3.4}, {4.2, 5.1, 6.1}, {7.2, 8.2, 9.1}}));
{λ,V}=Eigensystem[A]
(* Checking *)
i=1;
{λ[[i]] V[[i]], A.V[[i]]}