added lec 24

This commit is contained in:
Sasserisop 2023-11-04 16:36:36 -06:00
parent 42a20e4b22
commit bc3766c009
9 changed files with 1925 additions and 10 deletions

View File

@ -94,11 +94,11 @@ $A(x-y)=e^{y+2x}$
>$\lim_{ n \to 0 }\ln(n)=y+2x$ >$\lim_{ n \to 0 }\ln(n)=y+2x$
>$\lim_{ n \to 0 }\frac{d}{dx}\ln(n)=0=\frac{dy}{dx}+2$ >$\lim_{ n \to 0 }\frac{d}{dx}\ln(n)=0=\frac{dy}{dx}+2$
>$\frac{dy}{dx}=-2\quad \Box$ >$\frac{dy}{dx}=-2\quad \Box$
>so from $\frac{dy}{dx}=-2\frac{{x-y-\frac{1}{2}}}{x-y+1}$ we get: >plugging into the equation $(2x-2y-1)dx+(x-y+1)dy=0$ yields:
>$-2=-2\frac{{x-y-\frac{1}{2}}}{x-y+1}$ >$1=\frac{2(x-y+1)}{2x-2y-1}$
>$x-y+1=x-y-\frac{1}{2}$ >$2x-2y-1=2(x-y+1)$
>$1=-\frac{1}{2}$ >$-1=2$
>So what does this all mean? I honestly have no idea. I think it means we assumed that $e^{y+2x}=0$ is defined and because we arrived at a contradiction, our assumption was wrong. That didn't really get us to show if it was a valid solution or not like I imagined. >So what does this all mean? I think it means that even if we imagine that $\frac{dy}{dx}$ exists, the equation is not satisfied and $y=x$ is definitely not a solution even when we try to cheat a little.
We can rearrange to our liking, but we have found the general solution to the DE: We can rearrange to our liking, but we have found the general solution to the DE:
#### $$x-y=Ae^{2x+y}$$ #### $$x-y=Ae^{2x+y}$$

View File

@ -116,7 +116,7 @@ $\mathcal{L}^{-1}\{\alpha F(s)+\beta G(s)\}=\alpha \mathcal{L}^{-1}\{F\}+\beta \
This can be proven rather easily due to the linearity of the forward transform (wasn't done in class unfortunately). This can be proven rather easily due to the linearity of the forward transform (wasn't done in class unfortunately).
## Examples ## Examples
#ex #inv_LT I #ex #inv_LT
Compute this inverse LT: Compute this inverse LT:
$$\mathcal{L}^{-1}\left\{ \frac{1}{s^5}+\frac{3}{(2s+5)^2}+\frac{1}{s^2+4s+8}+ \frac{{s+1}}{s^2+2s+10} \right\}$$ $$\mathcal{L}^{-1}\left\{ \frac{1}{s^5}+\frac{3}{(2s+5)^2}+\frac{1}{s^2+4s+8}+ \frac{{s+1}}{s^2+2s+10} \right\}$$
Notice that all these terms approach 0 as s approaches inf. Notice that all these terms approach 0 as s approaches inf.

View File

@ -73,7 +73,7 @@ $\implies a_{n}=0$, $n=0,1,2,\dots$
#end of lec 22 #start of lec 23 #end of lec 22 #start of lec 23
Mid terms are almost done being marked! Mid terms are almost done being marked!
## Examples ## Solving DE using series
Let's start using power series to start solving DE! Let's start using power series to start solving DE!
No magic formulas we need to memorize when solving equations using power series (Yay!) No magic formulas we need to memorize when solving equations using power series (Yay!)
#ex #ex
@ -153,4 +153,47 @@ $z(x)=a_{0}\left( 1+\sum_{k=1}^\infty \frac{(1*4*\dots(3k-2))^2}{(3k)!}x^{3k} \r
$a_{1}\left( x+\sum_{k=1}^\infty \frac{(2\cdot 5\cdot \dots(3k-1))^2}{(3k+1)!} x^{3k+1}\right)$ $a_{1}\left( x+\sum_{k=1}^\infty \frac{(2\cdot 5\cdot \dots(3k-1))^2}{(3k+1)!} x^{3k+1}\right)$
there we go, $z$ is a linear combination of those two expressions there we go, $z$ is a linear combination of those two expressions
class done at 1:56 (a lil late but the journey is worth it) class done at 1:56 (a lil late but the journey is worth it)
#end of lec 23 #end of lec 23 #start of lec 24
*midterms have been marked and returned today.*
we consider:
$$y''+p(x)y'+q(x)y=0$$
this is in standard form, it's a second order linear equation
Definition:
if $p(x)$ and $q(x)$ are **analytic** functions in a vicinity of $x_{0}$ then $x_0$ is **ordinary**. Otherwise, $x_{0}$ is **singular**.
we expect that the solution y can be represented by a power series. This is true according to the following theorem:
Theorem: If $x_{0}$ is ordinary point then the differential equation above has two linearly independent solution of the form $\sum_{n=0} ^\infty a_{n}(x-x_{0})^n, \qquad\sum_{n=0}^\infty b_{n}(x-x_{0})^n$.
The radius of convergence for them is at least as large as the distance between $x_{0}$ and the closest singular point (which can be real or complex).
![[Drawing 2023-10-30 13.12.57.excalidraw.png]]
## Examples for calculating $\rho$
#ex
$$(x+1)y''-3xy'+2y=0 \quad x_{0}=1$$
put it in standard form:
$y''-\frac{3xy'}{x+1}+\frac{2y}{x+1}=0$
the only singular point for this equation is $x=-1$
so the minimum value of radius convergence is $\rho=2$ (distance between -1 and x_0)
we are guaranteed that the power series will converge *at least* in $(-1,3)$, possibly more. You can try solving for y as a power series.
#ex
$$y''-\tan xy'+y=0 \quad x_{0}=0$$
notice the coefficient beside y is 1, 1 is analytic and differentiable everywhere, obviously!
Same goes for any polynomial, it's obvious that any polynomial is infinitely differentiable but it's important to know.
What about tan x?
$\tan x=\frac{\sin x}{\cos x}$ is not defined on $x=\frac{\pi}{2}\pm n\pi, \qquad n=0,1,2,\dots$
the closest singular points are $\frac{\pi}{2}$ and $\frac{-\pi}{2}$ so our radius of convergence is the minimum distance of x_0 to these two points:
$\rho\geq\mid x_{0}-\frac{\pi}{2}\mid=\frac{\pi}{2}$
convergence could be larger, but we are guaranteed convergence on $x=x_{0}-\rho$ to $x_{0}+\rho$
#ex
$$(x^2+1)y''+xy'+y=0 \qquad x_{0}=1$$
put it in standard form:
$y''+\frac{x}{x^2+1}y'+\frac{y}{x^2+1}=0$
remember singular points can be complex the two singular points are:
$x^2=1=0 \qquad x=\pm i$
now we have to compute the two distances of these singular points to x=1
![[Drawing 2023-11-03 13.40.54.excalidraw.png]]
To calculate distance: $\alpha_{1}+\beta_{1}i, \qquad \alpha_{2}+\beta_{2}i$
$\sqrt{ (\alpha_{1}-\alpha_{2})^2+(\beta_{1}-\beta_{2})^2 }$
$\rho\geq \sqrt{ 1^2+1^2 }=\sqrt{ 2 }$
#end of lec 24

View File

@ -36,7 +36,7 @@ $\frac{d^2y}{dx^2}-y=xe^x$ <- notice the y here is not the same as the y above,
where $y(0)=0 \qquad \frac{dy}{dx}(0)=0$ where $y(0)=0 \qquad \frac{dy}{dx}(0)=0$
Hit it with the LT! Hit it with the LT!
$\frac{1}{s^2}$ is LT of $x$. Using the shifting property, $\frac{1}{(s-\alpha)^2}$ is the LT of $xe^{\alpha x}$ $\frac{1}{s^2}$ is LT of $x$. Using the shifting property, $\frac{1}{(s-\alpha)^2}$ is the LT of $xe^{\alpha x}$
$s^2Y(s)-Y(s) =\frac{1}{(s-1)^2}$ $s^2Y(s)-s\underbrace{ y(0) }_{ =0 }-\underbrace{ y'(0) }_{ =0 }-Y(s) =\frac{1}{(s-1)^2}$
Isolate $Y(s)$ : Isolate $Y(s)$ :
$(s^2-1)Y(s)=\frac{1}{(s-1)^2}$ $(s^2-1)Y(s)=\frac{1}{(s-1)^2}$
$Y(s)=\frac{1}{(s-1)^2(s^2-1)}$ $Y(s)=\frac{1}{(s-1)^2(s^2-1)}$
@ -45,6 +45,16 @@ Partial fraction time:
$Y(s)=\frac{1}{(s-1)^3(s+1)}=\frac{A}{s-1}+\frac{B}{(s-1)^2}+\frac{C}{(s-1)^3}+\frac{D}{s+1}$ $Y(s)=\frac{1}{(s-1)^3(s+1)}=\frac{A}{s-1}+\frac{B}{(s-1)^2}+\frac{C}{(s-1)^3}+\frac{D}{s+1}$
$\frac{{A(s-1)^2(s+1)+B(s-1)(s+1)+C(s+1)+D(s-1)^3}}{(s-1)^3(s+1)}$ $\frac{{A(s-1)^2(s+1)+B(s-1)(s+1)+C(s+1)+D(s-1)^3}}{(s-1)^3(s+1)}$
$\begin{matrix}A+D=0 \\A-2A+B-3D=0 \\ A-2A+B-B+C+3D=0 \\ A-B+C-D=1\end{matrix}$ $\begin{matrix}A+D=0 \\A-2A+B-3D=0 \\ A-2A+B-B+C+3D=0 \\ A-B+C-D=1\end{matrix}$
$\begin{bmatrix}1 & 0 & 0 & 1 & 0 \\-1 & 1 & 0 & -3 & 0 \\-1 & 0 & 1 & 3 & 0 \\ 1 & -1 & 1 & -1 & 1\end{bmatrix}$
$-A+B-3D=0$
$-A+C+3D=0$
$A=\frac{1}{8}$ $A=\frac{1}{8}$
$B=-\frac{1}{4}$ $B=-\frac{1}{4}$
$C=\frac{1}{2}$ $C=\frac{1}{2}$

View File

@ -22,7 +22,7 @@ Good luck on midterms! <3 -Oct 18 2023
[Periodic functions (lec 19)](periodic-functions-lec-19.html) (raw notes, not reviewed or revised yet.) [Periodic functions (lec 19)](periodic-functions-lec-19.html) (raw notes, not reviewed or revised yet.)
[Convolution (lec 19-20)](convolution-lec-19-20.html) (raw notes, not reviewed or revised yet.) [Convolution (lec 19-20)](convolution-lec-19-20.html) (raw notes, not reviewed or revised yet.)
[Dirak δ-function (lec 21)](dirak-δ-function-lec-21.html) (raw notes, not reviewed or revised yet.) [Dirak δ-function (lec 21)](dirak-δ-function-lec-21.html) (raw notes, not reviewed or revised yet.)
[Power series (lec 22-23)](power-series-lec-22-23.html) (raw notes, not reviewed or revised yet.) [Power series (lec 22-24)](power-series-lec-22-24.html) (raw notes, not reviewed or revised yet.)
</br> </br>
[How to solve any DE, a flow chart](Solve-any-DE.png) (Last updated Oct 1st, needs revision. But it gives a nice overview.) [How to solve any DE, a flow chart](Solve-any-DE.png) (Last updated Oct 1st, needs revision. But it gives a nice overview.)
[Big LT table (.png)](drawings/bigLTtable.png) [Big LT table (.png)](drawings/bigLTtable.png)

File diff suppressed because it is too large Load Diff

Binary file not shown.

After

Width:  |  Height:  |  Size: 20 KiB

View File

@ -0,0 +1,828 @@
---
excalidraw-plugin: parsed
tags: [excalidraw]
---
==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠==
# Text Elements
%%
# Drawing
```compressed-json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```
%%

Binary file not shown.

After

Width:  |  Height:  |  Size: 35 KiB