#import "custom.typ": mgrid, veq

#set heading(numbering:"1.")
#set math.equation(numbering: "(1)")

= Definitions <definitions>

order = the power the differential is raised to.
linear = the dependent variable and it's derivatives are all not non-linear.
$ mgrid(gutter: #1em,
    underbrace((dif ^2 y / dif t))
      & underbrace(cos(x) (dif y)/(dif x))
      & underbrace((dif y) / (dif t) (dif ^3 y) / (dif t^3))
      & underbrace(y^' = e^y)
      & underbrace(y (dif y)/(dif x)) \

    veq & veq & veq & veq \
  
    "linear" & "linear" & "non-linear" & "non-linear" & "non-linear" \
)$

autonomous = independent variable does not appear in the equation
non-autonomous = independent variable _does_ appear in the equation

ansatz = our initial guess for the form of a solution, i.e. $y_p = A cos (t) + B sin (t)$
indicial equation = a quadratic equation that pops out during the application of the Frobenius method

analytic = a function is analytic at a point if it can be expressed as a convergent power series in a neighborhood of that point  
ordinary point = when $p(x)$ and $q(x)$ are analytic at that point <ordinary-point>
regular singular point = if $P(x) = (x-x_0)p(x)$ and $Q(x) = (x-x_0)^2 q(x)$ are both analytic at $x_0$.
irregular singular point = not regular.

mean convergence = // todo
pointwise convergence =  // todo
uniform convergence = // todo

equilibrium point = // todo
stability // todo, this whole list
  - stable node = 
  - unstable bicritical node ("star") = 
  - stable centre = 
  - unstable saddle point =
  - unstable focus =

= Solving Methods

  == First Order
  table of contents

  === standard form
  $ (dif y) / (dif x) = f(x, y) $

  === separable
  $ (dif y)/(dif x) = f(x) g(y)
  arrow.r.double.long integral (dif y)/g(y) = integral f(x) dif x$

  === reduction to separable

  $ (dif y) / (dif x) = f ( y/x) $
  substitution: $y(x) = x v(x)$

  === linear standard form

  $ (dif y) / (dif x) + p(x) y = q(x) $ <linear-standard-form>

  ==== integrating factor

  note, the coefficient of $y'(x)$ must be 1.

  $ phi(x) = exp(integral p(x) dif x)$ <phi_x>

  multiplying the @linear-standard-form[Linear Standard Form] with $phi(x)$ yields:

  $ (dif)/(dif x)(phi y) = phi(x) q(x)
  arrow.r.double.long y = phi^(-1) integral phi q(x) dif x $

  === exact
  // todo

  == Second Order

  === standard form <second-order-standard-form>
  $ y^('') + p(x)y^' + q(x)y = r(x) $

  === reducible to first order

  $ (dif^2 y)/(dif x^2) + f(y, (dif y)/(dif x)) = 0$

  is reducible to the first-order ODE

  $ p (dif p)/(dif y) + f(y, p) = 0 $
  with substitution $p = (dif y)/(dif x)$

  === constant coefficients
  when $p(x)$ and $q(x)$ are constants:
  $ y^('') + a_1 y^' + a_0 y = 0 $

  ==== homogenous
  solve the characteristic equation:
  $ lambda^2 + a_1 lambda + a_0 = 0 $
  cases:
  - $lambda_1, lambda_2$ are real and distinct 
  - $lambda_1, lambda_2$ are real and coincide (same) 
  - $lambda_1, lambda_2$ are complex conjugates

  in each case, the solution of $y(x)$ becomes:
  - $y(x) = C exp(lambda_1 x) + D exp(lambda_2 x)$
  - $y(x) = C exp(lambda_1 x) + D x exp(lambda_1 x)$
  - $y(x) = C exp(alpha x)cos(beta x) + D exp(alpha x)sin(beta x) = exp(alpha x)(A cos(beta x) + B sin(beta x)) "by DeMoivre's Theorem" $

  ==== inhomogenous -> method of undetermined coefficients <method-uc>
  y(x) = y_h(x) + y_p(x)
  guesses for y_p(x):

  === variation of parameters
  This method works for any 2nd order inhomogenous ODE if the complementary solution is known.

  Theorem:
  The general solution of the 2nd order inhomogenous ODE:
  $ y^('') + b_1 (x) y^' + b_0 (x) y = f(x) $

  is given by $y(x) = u_1(x) y_1(x) + u_2(x) y_2(x)$
  
  where $y_1$ and $y_2$ are linearly independent solutions of the homogenous ODE such that the Wronskian $W(x) eq.not 0$ and 
  $ u_1(x) = -integral (y_2(x)f(x))/(W(x)) dif x, "and"
  u_2(x) = integral (y_1(x)f(x))/(W(x)) dif x $

  === power series method
  note, that we embark on this approach because the @second-order-standard-form[second order standard form] is not solveable in general with _elementary functions_!

  pick ansatz of the form
  $ y = sum^infinity_(n=0) a_n z^n $
  and take derivatives as required. for example:
  $ (dif y)/(dif z) = sum^infinity_(n=1) n a_n z^(n-1) 
  (dif^2 y)/(dif z^2) = sum^infinity_(n=2) n(n-1) a_n z^(n-2) $
  and substitute them into the ODE. Then solve by rearranging indices as necessary to obtain a recurrence relation. Apply the initial conditions and then guess the closed-form solution of the recurrence relation. Change back to the original variables if required.

  If $x_0$ is an ordinary point @definitions of the differential equation
  $ y^('') + p(x)y^' + q(x)y = 0 $
  then the general solution in a neighbourhood $| x - x_0 | < R$ may be represented as a power series.

  === method of frobenius
  Theorem:
  If $x_0 = 0$ is a regular singular point of the differential equation
  $ y^('') + p(x)y^' + q(x)y = 0 $
  then there exists at least one series solution of the form
  $ y(x) = x^r sum^infinity_(n=0) c_n x^n
  = sum^infinity_(n=0) c_n x^(n+r), c_0 eq.not 0$
  for some constant $r$ (index).

  ==== general indicial equation
  $ r(r-1) + p_0 r + q_0 = 0 $

  == n order
  admits $n$ linearly independent solutions.

  === power series expansion (not sure if it works for n order)
  // todo the general formula

  === reduction of order
  any $n^"th"$ order ODE can be formulated as a system of $n$ first order ODE's.

  // todo the formula

  == partial differential equations

  === standard form (linear, homogenous, 2nd order pde)
  $ A (partial^2 u)/(partial x^2) + B (partial^2 u)/(partial x partial y) + C (partial^2 u)/(partial y^2) + D(partial u)/(partial x) + E(partial u)/(partial y) + F u = 0 $

  parabolic equation: $B^2 - 4A C = 0$ @heat[Heat Equation]
  hyperbolic equation: $B^2 - 4A C > 0$ @wave[Wave Equation]
  elliptic equation: $B^2 - 4A C < 0$ @laplace-eqn[Laplace Equation]

  == separation of variables
  $ U(x,y) = X(x) Y(y) $
  then $U_x = Y X^'$ and $U_y = Y^' X$
  rewrite the PDE with these substitutions, then divide through by $X Y$. Integrate and solve.

  == change of variables
  // todo



= systems / dynamical systems

- $lambda_2 < lambda_1 < 0 arrow.r.double.long "stable node"$
- $0 < lambda_1 < lambda_2 arrow.r.double.long "unstable node"$
- $lambda_1 = lambda_2, lambda_1 > 0 arrow.r.double.long "unstable star"$
- $lambda_1 = lambda_2, lambda_1 < 0 arrow.r.double.long "stable star"$
- $lambda_1 < 0 < lambda_2 arrow.r.double.long "unstable saddle node"$
- $Re(lambda_1) = 0 arrow.r.double.long "centre, stable"$
- $Re(lambda_1) < 0 arrow.r.double.long "stable focus"$
- $Re(lambda_1) > 0 arrow.r.double.long "unstable focus"$

real canonical form // todo

= functions

== wronskian <wronskian>
$ "W"(f_1, f_2, ..., f_n)(x) = det(mat(
  f_1(x), f_2(x), ..., f_n(x);
  f_1'(x), f_2'(x), ..., f_n'(x);
  dots.v, dots.v, dots.down, dots.v;
  f_1^((n-1))(x), f_2^((n-1))(x), ..., f_n^((n-1))(x)
)) $
note that if a set of functions is linearly dependent, then its Wronskian will equal 0.

== power series, taylor series and maclaurin series expansions
#import "@preview/cetz:0.0.1": *

cetz.diagram(
  size: (20cm, 12cm),
  [
    ellipse[
      center: (10cm, 6cm),
      radius: (8cm, 4.5cm),
      fill: blue.lighten(4),
      stroke: blue,
      label: [top, "Power Series"],
    ],
    text[
      at: (10cm, 9.5cm),
      content: $sum_(n=0)^(infinity) a_n (x - a)^n$,
    ],

    ellipse[
      center: (10cm, 6cm),
      radius: (5.5cm, 3.2cm),
      fill: green.lighten(4),
      stroke: green,
      label: [top, "Taylor Series"],
    ],
    text[
      at: (10cm, 7.8cm),
      content: $sum_(n=0)^(infinity) (f^((n))(a))/(n!) (x - a)^n$,
    ],

    ellipse[
      center: (10cm, 6cm),
      radius: (3cm, 1.8cm),
      fill: orange.lighten(4),
      stroke: orange,
      label: [top, "Maclaurin Series"],
    ],
    text[
      at: (10cm, 6.8cm),
      content: $sum_(n=0)^(infinity) (f^((n))(0))/(n!) x^n$,
    ],
  ]
)

== orthogonality <orthogonal>
A set of functions ${phi_n}_(n=1,2,3,dots)$ is said to be orthogonal on the interval $[a,b]$ with respect to the inner product defined by
$ (f, g)_w = integral^b_a w(x)f(x)g(x) dif x $
with weight function $w(x) > 0$, if $(phi_n,phi_m)_w =0$ for $m eq.not n$.

== orthonormality <orthonormal>
a set ${phi_n}_(n=1,2,3,dots)$ is _orthonormal_ when in addition to being @orthogonal, $(phi_n,phi_n) = 1$, for $n = 1,2,3,dots.h$.

== cauchy-euler <cauchy-euler>
$ x^2 y^('') + a_1 x y^' + a_0 y = 0$
you can solve this by either letting $x = e^t$ or using the ansatz $y = x^lambda$
the characteristic equation is $lambda^2 + (a_1 - 1) lambda + a_0 = 0$
if you are blessed with the inhomogenous case of above, just use method of undetermined coefficients @method-uc.

== legendre <legendre>
legendre's (differential) equation
$ ( 1 - x^2 )y^('') - 2x y^' + n(n+1)y = 0 $
legendre's polynomials

== bessel <bessel>
bessel's differential equation
$ y^('') x^2 + x y^' + (x^2 - nu^2) y = 0 $

bessel function of the first kind of order $alpha$:

$ J_alpha(x) = sum^infinity_(m=0) (-1)^m / Gamma(m+1)Gamma(m+alpha+1) (x/2)^(2m+alpha) $
implies $ dif / dif x [x^alpha J_alpha(x)] = x^alpha J_(alpha-1)(x) $
implies $ integral^r_0 x^n J_(n-1) (x) dif x = r^n J_n(r) $ for $n = 1, 2, 3, dots.h$

thus the de admits solutions
case 1: $2 nu in.not ZZ $
$ y(x) = A J_nu(x) + B J_(-nu)(x) $
$J_nu(x)$,$J_(-nu)(x)$ linearly independent
case 2: $2 nu in ZZ $
$ y(x) = A J_nu(x) + B J_(-nu)(x) $
case 3: $nu in ZZ $
$J_nu(x)$,$J_(-nu)(x)$ linearly independent
$ y(x) = A J_nu(x) + B Y_(nu)(x) $

== laguerre's equation
<laguerre>
$ x y^('') + (1-x)y^' +n y = 0 $

== hermite's equation
<hermite>
$ y^('') - 2 x y^' + 2 n y = 0 $

== sturm-liouville form
$ (p y^')^' + (q + lambda r ) y = 0 $

note that @bessel, @laguerre, @hermite and @legendre equations can all be written in this form. furthermore, *any* 2nd order linear homogenous ODE $y^('') + a_1(x)y^' + [a_2(x) + lambda a_3(x)]y = 0$ may be written in this form.

== heat equation (pde) <heat>
$ (partial^2 u) / (partial x^2) = (partial u)/(partial t) $

== wave equation (pde) <wave>

$ (partial^2 u) / (partial x^2) = 1/c^2 (partial^2 u)/(partial t^2) $

== laplace's equation (pde ) <laplace-eqn>

$ (partial^2 u)/ (partial x^2) + (partial^2 u) / (partial y^2) = 0 $

== fourier series

$ y(x) = a_0/2 + sum^N_(n=1) (a_n cos (n x) + b_n sin (n x)) $
$ a_n = 1/pi integral^pi_(-pi) y(x) cos(n x) dif x, n = 0, 1, 2, dots.h$
$ b_n = 1/pi integral^pi_(-pi) y(x) sin(n x) dif x, n = 1, 2, dots.h$

parseval's identity
$ (|| f ||^2) / L = 1/L integral^L_(-L) f^2 dif x
= a_0 / 2 + sum^infinity_(n=1) (a_n^2 + b_n^2) $