diff --git a/content/Drawing 2023-10-11 13.17.32.excalidraw.md b/content/Drawing 2023-10-11 13.17.32.excalidraw.md new file mode 100644 index 0000000..530cda6 --- /dev/null +++ b/content/Drawing 2023-10-11 13.17.32.excalidraw.md @@ -0,0 +1,10616 @@ +--- + +excalidraw-plugin: parsed +tags: [excalidraw] + +--- +==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== + + +# Text Elements +%% +# Drawing +```json +{ + "type": "excalidraw", + "version": 2, + "source": "https://github.com/zsviczian/obsidian-excalidraw-plugin/releases/tag/1.9.19", + "elements": [ + { + "id": "npL0FgktyCa86l0yXVyui", + "type": "freedraw", + "x": -321.9468688964844, + "y": -209.6343994140625, + "width": 42, + "height": 45.5999755859375, + "angle": 0, + "strokeColor": "#1e1e1e", + "backgroundColor": "transparent", + "fillStyle": "hachure", + "strokeWidth": 1, + "strokeStyle": "solid", + "roughness": 1, + "opacity": 100, + "groupIds": [], + "frameId": null, + "roundness": null, + "seed": 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procudure always +# Reduction of order +$y''+p(x)y'+q(x)y=f(x)$ (1) <- no general solution procedure always but, if $y_{1}(x)$ solves $y''+p(x)y'+q(x)y=0$ then we can find the general solution to the non homogenous equation (1) by guessing it in the form $y(x)=v(x)y_{1}(x)$ $y'=v'y_{1}+vy_{1}'$ @@ -29,10 +29,10 @@ $v'=u$ $u'+\left( \frac{2y_{1}'}{y_{1}}+p \right)u=\frac{f}{y_{1}}$<- this is a linear first order equation how to solve linear first order equation? we compute the integrating factor $\mu$ $\mu=e^{\int(2y_{1}'/y_{1}+p)dx}=e^{\ln(y_{1})^2}e^{\int P(x) \, dx}=y_{1}^2e^{\int p(x) \, dx}$ -isnt this nice? some kind of magic. We made some guesses and we arrived somewhere. +Isn't this nice? some kind of magic. We made some guesses and we arrived somewhere. -#ex find the general solution to the equation: -$y''+4xy'+(4x^2+2)y=8e^{-x(x+2)}$ +#ex #reduction_of_order find the general solution to the equation: +$$y''+4xy'+(4x^2+2)y=8e^{-x(x+2)}$$ if $y_{1}(x)=e^{-x^2}$ is one solution. therefore were finding the solution of the form: $y(x)=v(x)y_{1}=v(x)e^{-x^2}$ $v'=u$ diff --git a/content/Laplace transform (lec 14).md b/content/Laplace transform (lec 14).md deleted file mode 100644 index 693c09b..0000000 --- a/content/Laplace transform (lec 14).md +++ /dev/null @@ -1,14 +0,0 @@ -# Laplace transform -From now on, LT is short for Laplace Transform -What is LT? It's denoted as $\mathcal{L}$ and it's an operator defined as the following: -$$\underset{ =F(s) }{ \mathcal{L}\{f(t)\}(s) }:=\int _{0}^\infty e^{-st}f(t) dt=\lim_{ T \to \infty }\int _{0} ^T e^{-st}f(t)\, dt$$ -This doesn't look like anything useful, but later on we will learn how it is. - -$\mathcal{L}\{0\}=0$ Look at your bank account, integrate 0 you still get 0 :D -$\mathcal{L}\{1\}=\int_{0}^\infty e^{-st} \, dt=-\frac{1}{s}e^{-st}|_{0}^\infty=\frac{1}{s}$ if s>0 -$\mathcal{L}\{e^{at}\}$=$\int_{0}^{\infty} e^{at}e^{-st}\, dt=\int_{0}^{\infty}e^{-(s-a)t}dt=\frac{1}{s-a}$ if $s-a>0$ -$\mathcal{L}\{\sin bt\}=\int _{0}^{\infty}e^{-st}\sin(bt) \, dt=\frac{b}{s^2+b^2}$ by integration by parts -similarly can be done for cos, but we have run out of time. -#end of lec 14 - - diff --git a/content/Laplace transform (lec 14-15).md b/content/Laplace transform (lec 14-15).md new file mode 100644 index 0000000..8521f11 --- /dev/null +++ b/content/Laplace transform (lec 14-15).md @@ -0,0 +1,63 @@ +# Laplace transform +From now on, LT is short for Laplace Transform +What is LT? It's denoted as $\mathcal{L}$ and it's an operator defined as the following: +$$\underset{ =F(s) }{ \mathcal{L}\{f(t)\}(s) }:=\int _{0}^\infty e^{-st}f(t) dt=\lim_{ T \to \infty }\int _{0} ^T e^{-st}f(t)\, dt$$ +This doesn't look like anything useful, but later on we will learn how it is. + +Here are some trivial example computations: +$\mathcal{L}\{0\}=0$ Look at your bank account, integrate 0 you still get 0 :D +$\mathcal{L}\{1\}=\int_{0}^\infty e^{-st} \, dt=-\frac{1}{s}e^{-st}|_{0}^\infty=\frac{1}{s}$ if s>0 +$\mathcal{L}\{e^{at}\}$=$\int_{0}^{\infty} e^{at}e^{-st}\, dt=\int_{0}^{\infty}e^{-(s-a)t}dt=\frac{1}{s-a}$ if $s-a>0$ +$\mathcal{L}\{\sin bt\}=\int _{0}^{\infty}e^{-st}\sin(bt) \, dt=\frac{b}{s^2+b^2}$ by integration by parts +similarly can be done for cos, but we have run out of time. +#end of lec 14 #start of lec 15 +compute the LT of this funny function: +$f(t)=\begin{pmatrix}1, & 0\leq t\leq 1 \\ 2, & 1T : f(t)\leq Me^{\alpha t}$ +this is important so that f(t)e^{-st} doesn't go into infinity, +this can be proven, but not necessary +Theorem: If $f(t)$ is piecewise continuous and of an exponential order $\alpha$ on $[0, \infty)$, then $\mathcal{L}\{f\}$ exists $s>\alpha$ +and we are guarenteed that $s>\alpha$ +$e^{\alpha t}$ is this of exponential order? Of course, $M=1,\alpha=\alpha$ +what about $\sin(t)$? yes, sin is bounded b/w -1 and 1 +What about $e^{t^2}$? No, t^2 always outgrows $\alpha t$ eventually. This function does not have a LT. + +![[Drawing 2023-10-11 13.21.18.excalidraw]] +Theorem if $F(s)$ is a LT then $\lim_{ s \to \infty }F(s)=0$ + +properties: +assume LT of f(t) exists: +first property: $\mathcal{L}\{e^{\alpha t}f(t)\}=\int _{0}^\infty e^{\alpha t}e^{-st}\, dt=\int _{0}^\infty e^{-(s-a)t}f(t)\, dt=F(s-a)$ + +$\mathcal{L}\{e^{\alpha t}\cdot 1\}=\frac{1}{s-\alpha}$ +$\mathcal{L}\{e^{\alpha t}\sin(bt)\}=\frac{b}{(s-a)^2+b^2}$ these properties are essential for the midterm. + +using integration by parts: +$\mathcal{L}\{f'(t)\}(s)=\int _{0}^\infty e^{-st}f(t)\, dt=e^{-st}f(t)|_{t=0}^{t\to \infty}+s \underbrace{ \int e^{-st}f(t) \, dt }_{ F(s) }$$=sF(s)-f(0)$ +$\mathcal{L}{f''}=s^2F(s)-sf(0)-f'(0))$ +using proof by induction (try at home!): +$\mathcal{L}\{f^{(m)}\}=s^mF(s)-s^{m-1}f(0)-\dots-f^{m-1}(0)$ + +trig stuff: +$\mathcal{L}\{\sin bt\}=\frac{b}{s^2+b^2}$ +$\mathcal{L}\{\cos bt\}=\frac{s}{s^2+b^2}$ +$\cos(bt)=\frac{1}{b}(\sin(bt))'$ + +$\frac{dF}{ds}(s)=\frac{d}{ds}\int _{0 } ^\infty e^{-st}f(t)\, dt=\int _{0} ^\infty f(t) \frac{d}{ds}(e^{-st}) \, dt$ +$=-\int _{0} ^\infty e^{-st}tf(t)\, dt$ +$-\frac{dF}{ds}=\mathcal{L}\{tf(t)\}$; again we can use induction to prove: +$\mathcal{L}\{t''f(t)\}=(-1)^{n} \frac{d^nF}{ds^n}$ +$\mathcal{L}\{t 1\}=-(s^-1)^1=s^-2$; $\mathcal{L}\{t^n\}= \frac{n!}{s^{n+1}}$ (can be proven by induction) +today covers all midterm material. Yay! +#end of lec 15 \ No newline at end of file diff --git a/content/Method of undetermined coefficients (lec 8-9).md b/content/Method of undetermined coefficients (lec 8-9).md index c188499..db86fd2 100644 --- a/content/Method of undetermined coefficients (lec 8-9).md +++ b/content/Method of undetermined coefficients (lec 8-9).md @@ -105,6 +105,6 @@ s=2 if r is a double root. case ii) $ay''+by'+cy=P_{m}(t)e^{\alpha t}\cos(\beta t)+Q_{n}(t)e^{\alpha t}\sin(\beta t)$ Then we guess the particular solution is of the form: $y_{p}(t)=t^s[(A_{k}t^k+A_{k-1}t^{k-1}+\dots+A_{0})e^{\alpha t}\cos(\beta t)+(B_{k}t^k+B_{k-1}t^{k-1}+\dots+B_{0})e^{\alpha t}\sin(\beta t)]$ where: -k=max(m,n) -s=0 if $\alpha+i\beta$ is not a root, -s=1 if $\alpha+i\beta$ is a root. \ No newline at end of file +$k=\max(m,n)$ +$s=0$ if $\alpha+i\beta$ is not a root, +$s=1$ if $\alpha+i\beta$ is a root. diff --git a/content/_index.md b/content/_index.md index e7f7409..72e17b8 100644 --- a/content/_index.md +++ b/content/_index.md @@ -13,7 +13,7 @@ I have written these notes for myself, I thought it would be cool to share them. [Cauchy-Euler equations (lec 10)](cauchy-euler-equations-lec-10.html) [Free vibrations (lec 11-12)](free-vibrations-lec-11-12.html) (raw notes, not reviewed or revised yet.) [Resonance & AM (lec 13-14)](resonance-am-lec-13-14.html) -[Laplace transform (lec 14)](laplace-transform-lec-14.html) +[Laplace transform (lec 14-15)](laplace-transform-lec-14-15.html) (raw notes, not reviewed or revised yet.)
[How to solve any DE, a flow chart](Solve-any-DE.png)