HW due next Monday
1.1 p.5 Read problems 1-14. Do even.
1.2 1(c), 2(a), 4, 11, 14, 17, 20, 22
1.1 and 1.2 Definitions

• What is a differential equation?
• Why do we work with differential equations?
• What is an ordinary differential equation?
• What is a partial differential equation?
• What is the order?
• What is a linear equation?
  In general, a linear ODE of n terms has the following form:
  a_n(x)(dⁿy/dxⁿ) + a_n-1(x)(d^n-1y/dx^n-1) + ... + a₀(x) = 0
  Examples of nonlinear equations
  dy/dx = y²
  dy/dx = siny
  y = x² nonlinear graph
  Examples of linear equations
  d²x/dt² = g
  dy/dx = y
  dy/dx = sinx
  y=3x + 2 linear graph
  My question--is the utility of an equation being linear (in solving differential equations) related to the order of the numerator because derivatives of any term (say y) will be of lesser order? What about y^-1 or ln y?
  
• What is a solution to an ODE? dy/dx = 2x
  An explicit solution is a function y = y(x) that satisfies the ODE.
  y' = 2x
  y = x² + C
  Several possible solutions. That is why we need an initial value.
  
• What is an initial value problem (IVP)? d²x/dt² = g = 9.8
  initial position x(0) = 10
  initial velocity x'(0) = .5
  

Laplace equation
partial derivatives
d²u/dx² + d²u/dy² = 0
heat equations
du/dt = d²u/dx² + d²u/dy²
u = u(x,t)
Second order PDE

Existance and uniqueness theorem

general idea:
An nth order ODE needs n initial values to pick up a unique solution.

P. 12 gives more complete theorem

Show that f(x) = x² - x^-1 is a solution to d²y/dx² - (2/(x²))y = 0
f'(x) = 2x + x^-2
f''(x) = 2 - 2(x^-3)
LHS = 2 - 2x^-3 - (2/(x²))(x² - x^-1)
(f(x) is substituted for y)
= 2x^-1/x²

Example
y'' - 3y' + 2y = 0
Find constant m, such that y = e^mx is a solution.

y' = me^mx
y'' = (m²)e^mx
(m²)e^mx - 3me^mx + 2e^mx = 0
m² - 3m + 2 = 0
(m - 1)(m - 2) = 0
m = 1 or m = 2
Therefore y = e^x or y=e^2x
should be solutions
y = e^x
y' = e^x
y'' = e^x
y = e^2x
y' = 2e^2x
y'' = 4e^2x