#maths #cheat_sheets

# ODEs
## constant  coefficient
[[wk 1 - ODEs#constant coefficient ODEs|more detail]]
### ansatz
$y=e^{\lambda x}$
### solutions
![[ODE solution table]]


## euler equations
[[wk 1 - ODEs#euler equations|more detail and examples]]
### ansatz
$y=x^n$
### solutions

real and distinct
- $y(x) = Ax^{n_1}+Bx^{n_2}$
 otherwise:
- look at the [[ODE solution table]] and use the substitution $x=\ln x$



## inhomogenous ODEs
### method
set right side to 0 and solve
solve the particular integral
add them together
done :)

### boundary value problems
[[wk 1 - ODEs#boundary conditions|more detail and examples]]

just substitute into the relevant amount of differentiating
they can be fully defined, infinite, or not have solutions




### guessing solutions for particular integrals
[[wk 1 - ODEs#inhomogeneous ODEs|more detail]]
|source term|guess|
|---|---|
|power of $x$ ($4x$)|that power of $x$ all the way down to $x^0$($Ax+b$)|
|trig function ($\sin 5x$)|$\sin$ and $\cos$ to that frequency ($B\sin 5x + C\cos 5x$)|
|$e$ to a power($25e^{3x})| $e$ to that power ($De^{3x}$)|
once a  guess is made, differentiate it enough times and then sub it into the left hand of the ODE. change coefficients so they are equal to the right hand of the ODE.


## eigenvalue problems
[[wk 1 - ODEs#eigenvalues|more detail (might not be great)]]
[[wk 5 - PDEs#lecture 2 - separating variables|may clarify]]
### the cases
1: $\lambda = 0$
2: $\lambda = k^2 > 0$
3: $\lambda = - k^2 < 0$
$k$ doesn't represent anything, and just comes from $\Lambda = \pm \sqrt{-\lambda}$

### solutions
1: $y=C_1+C_2x$, can sometimes be interesting
	$y' = C_2$
2: $y= C_1e^{kx} + C_2e^{-kx}$, usually not interesting
	$y'= k(C_1e^{kx} - C_2e^{-kx})$
3: $y=C_1\sin(kx) + C_2\cos(kx)$, almost always interesting
	$y'=k[C_1\cos(kx) - C_2\sin(kx)]$

# fourier series
![[wk 2 - intro to fourier series#trig identities]]
## euler formulae
in these, $\ell$ is the upper bound and $l$ is the lower bound
these are a lot simpler when the period is $2\pi$, ([[wk 2 - intro to fourier series#Euler formulae|link]])

$a_0 = \frac{1}{\ell} \int^\ell_l f(x) dx$
$a_n = \frac{1}{\ell} \int^\ell_l f(x) \cos\left(\frac{n\pi x}{\ell}\right) dx$
$b_n = \frac{1}{\ell} \int^\ell_l f(x) \sin\left(\frac{n\pi x}{\ell}\right) dx$

### even and odd functinos
[[wk 2 - intro to fourier series#even functions|read past just the highlighted bit]]
$\cos$ is even, $\sin$ is odd. These can simplify integrations.
if $f(x)$ is even, $b_n=0$ and it is a fourier cosine series
if $f(x)$ is odd, $a_n = 0$ and it is a fourier sine series

## integrating and differentiating fourier series
[[wk 3 - more fourier series and integral transforms#lecture 1|more detail]]
you can always integrate
differentiation isnt so easy

## complex fourier series
$\sin$ and $\cos$ can be written as $e$, so the fourier series can be written in terms of $e$

$c_n = \frac{1}{2\pi}\int^\ell_l f(x)e^{-\frac{n\pi x}{\ell}}dx$
$e^{j\pi n} = e^{-j\pi n}  =(-1)^n$
$e^{\pm j\pi} = -1$

$\sin(ax) = \frac{e^{jax}-e^{-jax}}{2j}$
$\cos(ax) = \frac{e^{jax}+e^{-jax}}{2}$


# fourier transforms
fourier transform: $\frac{1}{\sqrt{2\pi}} \int^\infty_{-\infty} f(t) e^{-j\omega t} dt$
inverse fourier transform: $\frac{1}{\sqrt{2\pi}} \int^\infty_{-\infty} F(\omega) e^{j\omega t} dt$
$\mathcal{F}[\frac{df}{dt}] = \bf j \omega \mathcal{F}[f(t)]$

# laplace transform
$\int^\infty_0 f(t) e^{-st}dt$

inverse laplace transforms
![[Pasted image 20230101150031.png|300]]
[[wk 4 - fourier and laplace transforms|last bit]] and [[wk 5 - PDEs|first bit]]

# PDEs

find boundary conditions ($y(0, t)$ and $y(L, t)$)

solve eigenvalue problem for $X''-\lambda X =0$