forked from Sasserisop/MATH201
added lec 24
This commit is contained in:
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@ -94,11 +94,11 @@ $A(x-y)=e^{y+2x}$
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>$\lim_{ n \to 0 }\ln(n)=y+2x$
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>$\lim_{ n \to 0 }\frac{d}{dx}\ln(n)=0=\frac{dy}{dx}+2$
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>$\frac{dy}{dx}=-2\quad \Box$
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>so from $\frac{dy}{dx}=-2\frac{{x-y-\frac{1}{2}}}{x-y+1}$ we get:
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>$-2=-2\frac{{x-y-\frac{1}{2}}}{x-y+1}$
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>$x-y+1=x-y-\frac{1}{2}$
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>$1=-\frac{1}{2}$
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>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.
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>plugging into the equation $(2x-2y-1)dx+(x-y+1)dy=0$ yields:
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>$1=\frac{2(x-y+1)}{2x-2y-1}$
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>$2x-2y-1=2(x-y+1)$
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>$-1=2$
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>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.
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We can rearrange to our liking, but we have found the general solution to the DE:
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#### $$x-y=Ae^{2x+y}$$
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@ -116,7 +116,7 @@ $\mathcal{L}^{-1}\{\alpha F(s)+\beta G(s)\}=\alpha \mathcal{L}^{-1}\{F\}+\beta \
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This can be proven rather easily due to the linearity of the forward transform (wasn't done in class unfortunately).
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## Examples
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#ex #inv_LT I
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#ex #inv_LT
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Compute this inverse LT:
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$$\mathcal{L}^{-1}\left\{ \frac{1}{s^5}+\frac{3}{(2s+5)^2}+\frac{1}{s^2+4s+8}+ \frac{{s+1}}{s^2+2s+10} \right\}$$
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Notice that all these terms approach 0 as s approaches inf.
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@ -73,7 +73,7 @@ $\implies a_{n}=0$, $n=0,1,2,\dots$
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#end of lec 22 #start of lec 23
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Mid terms are almost done being marked!
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## Examples
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## Solving DE using series
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Let's start using power series to start solving DE!
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No magic formulas we need to memorize when solving equations using power series (Yay!)
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#ex
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@ -153,4 +153,47 @@ $z(x)=a_{0}\left( 1+\sum_{k=1}^\infty \frac{(1*4*\dots(3k-2))^2}{(3k)!}x^{3k} \r
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$a_{1}\left( x+\sum_{k=1}^\infty \frac{(2\cdot 5\cdot \dots(3k-1))^2}{(3k+1)!} x^{3k+1}\right)$
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there we go, $z$ is a linear combination of those two expressions
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class done at 1:56 (a lil late but the journey is worth it)
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#end of lec 23
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#end of lec 23 #start of lec 24
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*midterms have been marked and returned today.*
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we consider:
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$$y''+p(x)y'+q(x)y=0$$
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this is in standard form, it's a second order linear equation
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Definition:
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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**.
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we expect that the solution y can be represented by a power series. This is true according to the following theorem:
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Theorem: If $x_{0}$ is ordinary point then the differential equation above has two linearly independent solution of the form $\sum_{n=0} ^\infty a_{n}(x-x_{0})^n, \qquad\sum_{n=0}^\infty b_{n}(x-x_{0})^n$.
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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).
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![[Drawing 2023-10-30 13.12.57.excalidraw.png]]
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## Examples for calculating $\rho$
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#ex
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$$(x+1)y''-3xy'+2y=0 \quad x_{0}=1$$
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put it in standard form:
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$y''-\frac{3xy'}{x+1}+\frac{2y}{x+1}=0$
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the only singular point for this equation is $x=-1$
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so the minimum value of radius convergence is $\rho=2$ (distance between -1 and x_0)
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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.
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#ex
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$$y''-\tan xy'+y=0 \quad x_{0}=0$$
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notice the coefficient beside y is 1, 1 is analytic and differentiable everywhere, obviously!
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Same goes for any polynomial, it's obvious that any polynomial is infinitely differentiable but it's important to know.
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What about tan x?
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$\tan x=\frac{\sin x}{\cos x}$ is not defined on $x=\frac{\pi}{2}\pm n\pi, \qquad n=0,1,2,\dots$
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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:
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$\rho\geq\mid x_{0}-\frac{\pi}{2}\mid=\frac{\pi}{2}$
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convergence could be larger, but we are guaranteed convergence on $x=x_{0}-\rho$ to $x_{0}+\rho$
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#ex
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$$(x^2+1)y''+xy'+y=0 \qquad x_{0}=1$$
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put it in standard form:
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$y''+\frac{x}{x^2+1}y'+\frac{y}{x^2+1}=0$
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remember singular points can be complex the two singular points are:
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$x^2=1=0 \qquad x=\pm i$
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now we have to compute the two distances of these singular points to x=1
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![[Drawing 2023-11-03 13.40.54.excalidraw.png]]
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To calculate distance: $\alpha_{1}+\beta_{1}i, \qquad \alpha_{2}+\beta_{2}i$
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$\sqrt{ (\alpha_{1}-\alpha_{2})^2+(\beta_{1}-\beta_{2})^2 }$
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$\rho\geq \sqrt{ 1^2+1^2 }=\sqrt{ 2 }$
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#end of lec 24
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@ -36,7 +36,7 @@ $\frac{d^2y}{dx^2}-y=xe^x$ <- notice the y here is not the same as the y above,
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where $y(0)=0 \qquad \frac{dy}{dx}(0)=0$
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Hit it with the LT!
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$\frac{1}{s^2}$ is LT of $x$. Using the shifting property, $\frac{1}{(s-\alpha)^2}$ is the LT of $xe^{\alpha x}$
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$s^2Y(s)-Y(s) =\frac{1}{(s-1)^2}$
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$s^2Y(s)-s\underbrace{ y(0) }_{ =0 }-\underbrace{ y'(0) }_{ =0 }-Y(s) =\frac{1}{(s-1)^2}$
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Isolate $Y(s)$ :
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$(s^2-1)Y(s)=\frac{1}{(s-1)^2}$
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$Y(s)=\frac{1}{(s-1)^2(s^2-1)}$
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@ -45,6 +45,16 @@ Partial fraction time:
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$Y(s)=\frac{1}{(s-1)^3(s+1)}=\frac{A}{s-1}+\frac{B}{(s-1)^2}+\frac{C}{(s-1)^3}+\frac{D}{s+1}$
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$\frac{{A(s-1)^2(s+1)+B(s-1)(s+1)+C(s+1)+D(s-1)^3}}{(s-1)^3(s+1)}$
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$\begin{matrix}A+D=0 \\A-2A+B-3D=0 \\ A-2A+B-B+C+3D=0 \\ A-B+C-D=1\end{matrix}$
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$\begin{bmatrix}1 & 0 & 0 & 1 & 0 \\-1 & 1 & 0 & -3 & 0 \\-1 & 0 & 1 & 3 & 0 \\ 1 & -1 & 1 & -1 & 1\end{bmatrix}$
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$-A+B-3D=0$
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$-A+C+3D=0$
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$A=\frac{1}{8}$
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$B=-\frac{1}{4}$
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$C=\frac{1}{2}$
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@ -22,7 +22,7 @@ Good luck on midterms! <3 -Oct 18 2023
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[Periodic functions (lec 19)](periodic-functions-lec-19.html) (raw notes, not reviewed or revised yet.)
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[Convolution (lec 19-20)](convolution-lec-19-20.html) (raw notes, not reviewed or revised yet.)
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[Dirak δ-function (lec 21)](dirak-δ-function-lec-21.html) (raw notes, not reviewed or revised yet.)
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[Power series (lec 22-23)](power-series-lec-22-23.html) (raw notes, not reviewed or revised yet.)
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[Power series (lec 22-24)](power-series-lec-22-24.html) (raw notes, not reviewed or revised yet.)
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</br>
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[How to solve any DE, a flow chart](Solve-any-DE.png) (Last updated Oct 1st, needs revision. But it gives a nice overview.)
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[Big LT table (.png)](drawings/bigLTtable.png)
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@ -0,0 +1,828 @@
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---
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excalidraw-plugin: parsed
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tags: [excalidraw]
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---
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==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠==
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# Text Elements
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%%
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# Drawing
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||||
```
|
||||
%%
|
||||
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|
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Loading…
Reference in New Issue