diff --git a/content/Drawing 2023-10-06 13.24.11.excalidraw.md b/content/Drawing 2023-10-06 13.24.11.excalidraw.md new file mode 100644 index 0000000..38ec524 --- /dev/null +++ b/content/Drawing 2023-10-06 13.24.11.excalidraw.md @@ -0,0 +1,20534 @@ +--- + +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": "4PlMDegOfa-FhsAMQQpQY", + "type": "freedraw", + "x": -351.45001220703125, + "y": -299.6343994140625, + "width": 294, + "height": 14.999984741210938, + "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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RHS=0. $mr^2+br+k=0$ characteristic polynomial (i) $r_{1}\ne r_{2}$ $b^2-4mk>0$ $y_{h}(t)=c_{1}e^{r_{1}t}+c_{2}e^{r_{2}t}$ diff --git a/content/Laplace transform (lec 14).md b/content/Laplace transform (lec 14).md new file mode 100644 index 0000000..693c09b --- /dev/null +++ b/content/Laplace transform (lec 14).md @@ -0,0 +1,14 @@ +# 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/Method of undetermined coefficients (lec 8-9).md b/content/Method of undetermined coefficients (lec 8-9).md index 33c2d80..fa6a019 100644 --- a/content/Method of undetermined coefficients (lec 8-9).md +++ b/content/Method of undetermined coefficients (lec 8-9).md @@ -67,9 +67,9 @@ We wanna solve this IVP! We know from earlier that it must have a unique solutio $y_{p}(t)=y_{p_{1}}(t)+y_{p_{2}}(t)$ where $y_{p_{1}}$ solves $y''+2y'+2y=2e^{-t}$ $y_{p_{2}}$ solves $y''+2y'+2y=5\cos (t)$ - lets try $y_{p_{1}}=Ae^{-t}$ Does this work? look at it, A must be zero but if A is zero you still get problems. + lets try $y_{p_{1}}=Ae^{-t}$ Does this work? $y_{p_{1}}'=-Ae^{-t}$ - $y_{p_{1}}''=Ae^{-t}$ plug in these three and we find that A=2 + $y_{p_{1}}''=Ae^{-t}$ plug in these three and we find that A=2. Yes it works! second equation, not so easy: solution of $\cos(t)$ doesn't quite work because the LHS will obtain a $\sin$ term. Lets try this instead: diff --git a/content/Prof variation of parameters.md b/content/Prof variation of parameters.md new file mode 100644 index 0000000..ae6f09b --- /dev/null +++ b/content/Prof variation of parameters.md @@ -0,0 +1,22 @@ + +# Variation of parameters +$ay''+by'+cy=f(t)$ +1) $y_{h}=c_{1}y_{1}(t)+c_{2}y_{2}(t)$ <- h is homogenous, ie: $f(t)=0$ +Lagrange proposed this method to find the particular solution $y_{p}$: +$y_{p}(t)=v_{1}(t)y_{1}(t)+v_{2}(t)y_{2}(t)$ <- btw $y_{1}$ and $y_{2}$ are often called a fundamental pair. +we put $y_p$ into the equation and make it equal to the RHS. To do so, find the derivatives first: +$y'_{p}=v_{1}'y_{1}+v_{1}y_{1}'+v_{2}'y_{2}+v_{2}y_{2}'$ +to avoid second derivatives in the equation and problems with uniqueness, Lagrange imposed: +1) $v_{1}'y_{1}+v_{2}'y_{2}=0$ this simplifies our work down the road as well. +$y'_{p}=\cancel{ v_{1}'y_{1} }+v_{1}y_{1}'+\cancel{ v_{2}'y_{2} }+v_{2}y_{2}'$ +so $y_{p}''=v_{1}'y_{1}'+v_{1}y_{1}''+v_{2}'y_{2}'+v_{2}y_{2}''$ +now we plug into the second order equation: +$a(v_{1}'y_{1}'+v_{1}y_{1}''+v_{2}'y_{2}'+v_{2}y_{2}'')+b(v_{1}y_{1}'+v_{2}y_{2}')+c(v_{1}y_{1}+v_{2}y_{2})=f(t)$ +$v_{1}(\cancel{ ay_{1}''+ by_{1}' +cy_{1} })+v_{2}(\cancel{ ay_{2}''+ by_{2}'+cy_{2} })+a(v_{1}'y_{1}'+v_{2}'y_{2}')$ +Both $y_{1}$ and $y_{2}$ are solutions to the homogenous counterpart. So the first two terms above equal to zero. +2) $v_{1}'y_{1}'+v_{2}'y_{2}'=\frac{f(t)}{a}$ +we now have a system of two equations: +$\det \begin{pmatrix}y_{1} & y_{2} \\y_{1}'& y_{2}'\end{pmatrix}$ = Wronskian = $W[y_{1},y_{2}]\ne 0$ +by definition $y_1$ and $y_2$ are linearly independent solutions so the above can never be 0! +$v_{1}'=-\frac{{f(t)y_{2}t}}{aW[y_{1},y_{2}]}$; $v_{2}'=\frac{{f(t)y_{1}(t)}}{aW[y_{1},y_{2}]}$ <- integrate both sizes to get $v_{1}$ and $v_{2}$. When integrating, you don't need to add a generic constant. +Finally, your solution is: $y_{p}(t)=v_{1}(t)y_{1}(t)+v_{2}(t)y_{2}(t)$ where you get $y_{1},y_{2}$ from your homogenous solution. diff --git a/content/Resonance & AM (lec 13-14).md b/content/Resonance & AM (lec 13-14).md new file mode 100644 index 0000000..7daa9fd --- /dev/null +++ b/content/Resonance & AM (lec 13-14).md @@ -0,0 +1,83 @@ +#start of lec 13 +He has good news. he's excited to tell us about electric currents! In particular, how a radio uses resonance to selectively listen to a particular frequency: +# Resonance +Let's imagine a mass spring system which has an applied force with a forcing frequency of $\gamma$ and an amplitude of $F_{o}$ (a constant): +$my''+by'+ky=F_{o}\cos(\gamma t)$ +(Think of the driving force being the radio transmitter, and the mass-spring system is an LC tank circuit in a old-style radio) +In order to study the phenomenon of resonance, we need an underdamped system. +so we let: $b^2-4mk<0$ (ie: complex roots) +then the homogenous solution becomes: +$y_{h}(t)=e^{-bt/2m}\left( c_{1}\cos\left( \frac{\sqrt{ 4mk-b^2 }}{2m} t\right)+c_{2}\sin\left( \frac{\sqrt{ 4mk-b^2 }}{2m} t\right) \right)$ +$y_{h}(t)=Ae^{-bt/2m}\sin(\omega t+\phi)$ +where $\omega$ is called the angular frequency and equals $\frac{\sqrt{ 4mk-b^2 }}{2m}$ +and where $Ae^{-bt/2m}$ is called the transient part of the equation (goes to 0 as t->$\infty$). +For particular solution, we use method of undetermined coefficients #mouc +We guess: $y_p=A_{1}\cos(\gamma t)+A_{2}\sin(\gamma t)$ where $\gamma\ne \omega$ because if $\gamma=\omega$ and b=0 then we would have to multiply our guess by t. +$A_{1}=\frac{F_{o}(k-m\gamma)}{(k-m\gamma^2)^2+b^2\gamma^2}$ +$A_{2}=\frac{F_{o}b\gamma}{(k-m\gamma^2)^2+b^2\gamma^2}$ +$y_{p}(t)=\frac{F_{o}}{(k-m\gamma^2)^2+b^2\gamma^2}((k-m\gamma^2 )\cos(\gamma t)+b\gamma \sin \gamma t)$ +$\sin(\alpha+\beta)=\sin \alpha \cos \beta+\cos \alpha \sin \beta$ +$k-m\gamma^2=A\sin \theta$ +$br=A\cos \theta$ +$A=\sqrt{ (k-m\gamma^2)^2+b^2\gamma^2 }$ +$\tan \theta=\frac{k-m\gamma^2}{b\gamma}$ +$y_{p}(t)=\frac{F_{o}}{(k-m\gamma^2)^2+b^2\gamma^2}((k-m\gamma^2 )\cos(\gamma t)+b\gamma \sin \gamma t)=\frac{F_{o}\sqrt{ (k-m\gamma^2)^2+b^2\gamma^2 }}{(k-m\gamma^2)^2+b^2\gamma^2}\sin(\gamma t+\theta)$ +$=\frac{F_{o}}{\sqrt{ (k-m\gamma^2)^2+b^2\gamma^2 }}\sin(\gamma t+\theta)$ +where we define $\mu(\gamma)=\frac{1}{\sqrt{ (k-m\gamma^2)^2+b^2\gamma^2 }}$ called the gain factor +and the general solution is $y(t)=y_{h}(t)+y_{p}(t)$ + +See how $y_h$ goes to zero as time progresses but $y_p$ stays? $y_p$ is the steady state part of the solution. If you were to graph $y(t)$ you would see a "beating" effect due to the sum of the two sins that eventually decays off. + +If we make the value in the denominator of the gain factor small, the amplitude goes to a very high value, higher than $F_{o}$! This is the equivalent of tuning a radio receiving circuit to resonate to a certain transmitted frequency, the amplitude grows to a great value in the receiver when excited with the right frequency. This is resonance! as b (friction) is decreased to zero we get stronger and stronger resonance. + +Lets find the maximum of the amplitude (resonance point) +take the derivative of $\mu$ wrt to $\gamma$: +$\mu'(\gamma)=-\frac{{2m^2\gamma\left( \gamma^2-\left( \frac{k}{m}-\frac{b^2}{2m^2} \right) \right)}}{[(k-m\gamma^2)^2+b^2\gamma^2]^{3/2}}=0$ +case one: $\gamma=0$ not interesting, because then the force applied would just a constant force. +case two: +$\gamma_{r}=\sqrt{ \frac{k}{m}-\frac{b^2}{2m^2} }$ where the r means resonance +By plugging in $\gamma_{r}$ into $\mu()$ we can know the maximum amplitude at resonance: $\mu_{max}(\gamma_{r})=\mu(\gamma_{r})=\frac{2m}{b\sqrt{ 4mk-b^2 }}$ +if $b^2\geq 4mk>2mk$ (overly damped/critically damped case) (no resonance, imaginary numbers) $\gamma=0$ +if $b^2<2mk<4mk$ (assumed from the very beginning above) then we get a resonant frequency: $\gamma_{r}=\sqrt{\frac{ 2mk-b^2}{2m^2}}$ + +what if b=0? (no resistance): +$my''+ky=F_{o}\cos(\gamma t)$ +$y_{h}(t)=c_{1}\cos \omega t+c_{2}\sin \omega t$ , $\omega=\sqrt{ \frac{k}{m} }$ +$=A\sin(\omega t+\phi)$ + +$y_{p}(t)=A_{1}\cos(\gamma t)+A_{2}\sin(\gamma t)$ +assume $\gamma=\omega$ with zero resistance we get: +$y_{p}(t)=\frac{F_{o}}{2m\omega}t\sin \omega t \underset{ t\to \infty }{ \to }\infty$ (your circuit blows up! Or equivalently, your bridge collapses.) +#end of lec 13 +#start of lecture 14 +# Amplitude modulation +Last lecture we showed we can selectively listen to a specific signal by using resonance. +He has something else he's excited to show us; amplitude modulation! Lets recall from last lecture: +$my''+ky=F_{o}\cos(\gamma t)$ (undamped driven mass-spring system) +solving characteristic equation: +$\omega=\sqrt{ \frac{k}{m} }$ +$y_{h}(t)=c_{1}\cos(\omega t)+c_{2}\sin(\omega t)$ ; $\gamma\ne\omega$ +$y_{p}=A\cos(\gamma t)$ we can guess the particular solution is a constant times $\cos$. There will be no $\sin$ term on the LHS as there's no first derivative (aka no friction) +$\Rightarrow A=\frac{F_{o}}{m(\omega^2-\gamma^2)}\cos(\gamma t)$ +$y(t)=c_{1}\cos(\omega t)+c_{2}\sin(\omega t)+\frac{F_{o}}{m(\omega^2-\gamma^2)}\cos(\gamma t)$ +assume $y(0)=0=y'(0)$ +$c_{2}=0$ $c_{1}=-\frac{F_{o}}{m(\omega^2-\gamma)}$ +$y(t)=\frac{F_{o}}{m(\omega^2-\gamma^2)}(\cos(\gamma t)-\cos(\omega t))$ +Hmm, It's not very easy to visualize a cos minus cos term. +Use trig identity to make equation easier to visualize: $2\sin(\alpha)\sin(\beta)=\cos\left( \frac{{\alpha-\beta}}{2} \right)-\cos\left( \frac{{\alpha+\beta}}{2} \right)$ +$2\gamma t=\alpha-\beta$ +$2\omega t=\alpha+\beta$ +$\Rightarrow \alpha=\frac{(\omega+\gamma)t}{2}$ $\beta=\frac{(\omega-\gamma)t}{2}$ +$$y(t)=\frac{F_{o}}{m(\omega^2-\gamma^2)}\sin\frac{(\omega-\gamma)t}{2}\sin\frac{(\omega+\gamma)t}{2}$$ +This is an amplitude modulated signal! also can be seen as a "beating frequency". + +To recap, taking a frictionless mass spring system which is initially at rest, and applying a driving frequency that is strictly different from the system's natural frequency, results in a displacement that follows an AM modulated signal. How cool is that! +>Sometimes I feel this effect when I'm sitting in the back of the bus, where the bus is stopped at a red light. I wonder if this modulated vibration pattern is attributed to this exact phenomenon. + +# Shortcut for solving DE of a mass spring system +![[Drawing 2023-10-06 13.24.11.excalidraw]] +$my''+by'+ky=mg+F$ +move $mg$ to LHS and replace $y$ with $y_{new}$ (remember, the $\frac{mg}{k}$ is a constant, its derivative is 0): +$m\left( y-\frac{mg}{k} \right)''+b\left( y-\frac{mg}{k} \right)'+k\left( y-\frac{mg}{k} \right)=F$ +$my_{new}''+by_{new}'+ky_{new}=F$ +This simplifies our approach to solving a mass spring system. We could do it without this rearrangement, but it's more complex as the RHS has a sum of two terms. Either way works though, pick what you like. \ No newline at end of file diff --git a/content/Resonance in free vibrations (lec 13).md b/content/Resonance in free vibrations (lec 13).md deleted file mode 100644 index 062991d..0000000 --- a/content/Resonance in free vibrations (lec 13).md +++ /dev/null @@ -1,41 +0,0 @@ -#start of lec 13 -He has good news. he's excited to tell us about electric currents! -$my''+by'+ky=F_{o}\cos(\gamma t)$ the system has an applied force with a forcing frequency of $\gamma$ and an amplitude of $F_{o}$ (a constant) -$b^2-4mk<0$ (complex roots) -$y_{h}(t)=e^{-bt/2m}\left( c_{1}\cos\left( \frac{\sqrt{ 4mk-b^2 }}{2m} t\right)+c_{2}\sin\left( \frac{\sqrt{ 4mk-b^2 }}{2m} t\right) \right)$ -$y_{h}(t)=Ae^{-bt/2m}\sin(\omega t+\phi)$ where $\omega$ is angular frequency and equals $\frac{\sqrt{ 4mk-b^2 }}{2m}$ -where $Ae^{-bt/2m}$ is called the transient part of the equation (makes equation go to 0 as t->$\infty$) -We guess: $y_p=A_{1}\cos(\gamma t)+A_{2}\sin(\gamma t)$ where $\gamma\ne \omega$ because if $\gamma=\omega$ and b=0 then we would have to multiply our guess by t. -$A_{1}=\frac{F_{o}(k-m\gamma)}{(k-m\gamma^2)^2+b^2\gamma^2}$ -$A_{2}=\frac{F_{o}b\gamma}{(k-m\gamma^2)^2+b^2\gamma^2}$ -$y_{p}(t)=\frac{F_{o}}{(k-m\gamma^2)^2+b^2\gamma^2}((k-m\gamma^2 )\cos(\gamma t)+b\gamma \sin \gamma t)$ -$\sin(\alpha+\beta)=\sin \alpha \cos \beta+\cos \alpha \sin \beta$ -$k-m\gamma^2=A\sin \theta$ -$br=A\cos \theta$ -$A=\sqrt{ (k-m\gamma^2)^2+b^2\gamma^2 }$ -$\tan \theta=\frac{k-m\gamma^2}{b\gamma}$ -$y_{p}(t)=\frac{F_{o}}{(k-m\gamma^2)^2+b^2\gamma^2}((k-m\gamma^2 )\cos(\gamma t)+b\gamma \sin \gamma t)=\frac{F_{o}\sqrt{ (k-m\gamma^2)^2+b^2\gamma^2 }}{(k-m\gamma^2)^2+\beta^2\gamma^2}\sin(\gamma t+\theta)$ -$=\frac{F_{o}}{\sqrt{ (k-m\gamma^2)^2+b^2\gamma^2 }}\sin(\gamma t+\theta)$ -where we define $\mu(\gamma)=\frac{1}{(k-m\gamma^2)^2+b^2\gamma^2}$ called the gain factor -and the general solution is $y(t)=y_{h}(t)+y_{p}(t)$ - -see how y_h goes to zero but y_p stays? y_p is the steady state part of the solution. if you were to graph it you would see a wah wah effect that decays to zero. - -if we make the value in the denominator of the gain factor small the amplitude goes to a very high value, higher than $F_{o}$ this is equivalent to tuning a radio circuit to resonate to a certain transmitted frequency, the amplitude grows to a great value in the receiver when excited with the right frequency. This is resonance! as b (friction) decreases to zero we get stronger and stronger resonance. -lets find the maximum of the amplitude (resonance point) -$\mu'(\gamma)=-\frac{{2m^2\gamma\left( \gamma^2-\left( \frac{k}{m}-\frac{b^2}{2m^2} \right) \right)}}{[(k-m\gamma^2)^2+b^2\gamma^2]^{3/2}}=0$ -cases: $\gamma=0$ not interesting, beacuse then the force applied is a constant force. -$\gamma_{r}=\sqrt{ \frac{k}{m}-\frac{b^2}{2m^2} }$ where the r means resonance -by taking second derivative$\mu_{max}(\gamma_{r})=\frac{2m}{b\sqrt{ 4mk-b^2 }}$ -if $b^2\geq 4mk>2mk$ (overly damped/critically damped case) (no resonance, imaginary numbers) $\gamma=0$ -if $b^2<2mk<4mk$ (assumed from the beginning above) then we get a resonant frequency $\gamma_{r}=\sqrt{\frac{ 2mk-b^2}{2m^2}}$ - -what if b=0? (no resistance): -$my''+ky=F_{o}\cos(\gamma t)$ -$y_{h}(t)=c_{1}\cos \omega t+c_{2}\sin \omega t$ , $\omega=\sqrt{ \frac{k}{m} }$ -$=A\sin(\omega t+\phi)$ - -$y_{p}(t)=A_{1}\cos(\gamma t)+A_{2}\sin(\gamma t)$ -assume $\gamma=\omega$ with zero resistance we get: -$y_{p}(t)=\frac{F_{o}}{2m\omega}t\sin \omega t \underset{ t\to \infty }{ \to }\infty$ (your circuit blows up! Or equivalently, your bridge collapses.) -#end of lec 13 \ No newline at end of file diff --git a/content/Variation of parameters (lec 9-10).md b/content/Variation of parameters (lec 9-10).md index c1e2eeb..ef19ae0 100644 --- a/content/Variation of parameters (lec 9-10).md +++ b/content/Variation of parameters (lec 9-10).md @@ -1,90 +1,105 @@ -# Variation of parameters -$ay''+by'+cy=f(t)$ -1) $y_{h}=c_{1}y_{1}(t)+c_{2}y_{2}t$ <- h is homogenous, ie: $f(t)=0$ -Lagrange proposed: find a particular solution of $y_{p}$ -$y_{p}(t)=v_{1}(t)y_{1}(t)+v_{2}(t)y_{2}(t)$ <- btw $y_{1}$ and $y_{2}$ are often called a fundamental pair. -we put $y_p$ into the equation and make it equal to the RHS. To do so, find the derivatives first: -$y'_{p}=v_{1}'y_{1}+v_{1}y_{1}'+v_{2}'y_{2}+v_{2}y_{2}'$ -to avoid second derivatives in the equation and problems with uniqueness, Lagrange imposed: -1) $v_{1}'y_{1}+v_{2}'y_{2}=0$ this simplifies our work down the road as well. -$y'_{p}=\cancel{ v_{1}'y_{1} }+v_{1}y_{1}'+\cancel{ v_{2}'y_{2} }+v_{2}y_{2}'$ -so $y_{p}''=v_{1}'y_{1}'+v_{1}y_{1}''+v_{2}'y_{2}'+v_{2}y_{2}''$ -now we plug into the second order equation: -$a(v_{1}'y_{1}'+v_{1}y_{1}''+v_{2}'y_{2}'+v_{2}y_{2}'')+b(v_{1}y_{1}'+v_{2}y_{2}')+c(v_{1}y_{1}+v_{2}y_{2})=f(t)$ -$v_{1}(\cancel{ ay_{1}''+ by_{1}' +cy_{1} })+v_{2}(\cancel{ ay_{2}''+ by_{2}'+cy_{2} })+a(v_{1}'y_{1}'+v_{2}'y_{2}')$ -Both $y_{1}$ and $y_{2}$ are solutions to the homogenous counterpart. So the first two terms above equal to zero. -2) $v_{1}'y_{1}'+v_{2}'y_{2}'=\frac{f(t)}{a}$ -$\det \begin{pmatrix}y_{1} & y_{2} \\y_{1}'& y_{2}'\end{pmatrix}$ = Wronskian = $W[y_{1},y_{2}]\ne 0$ -by definition $y_1$ and $y_2$ are linearly independent solutions so the above can never be 0! -$v_{1}'=\frac{{f(t)y_{2}t}}{aW[y_{1},y_{2}]}$; $v_{2}'=-\frac{{f(t)y_{1}(t)}}{aW[y_{1},y_{2}]}$ <- integrate both sizes to get $v_{1}$ and $v_{2}$. When integrating, you don't need to add a generic constant. +# Variation of parameters +#voparam +>[Professors definition/derivations during lecture](prof-variation-of-parameters.html) <- I found this to be too big brain for me. Here is a simplified definition: -#ex #second_order #IVP +## Variation of parameters is a method to solve: +## $$ay''+by'+cy=f(t)$$ +First, find the homogenous solution: +$y_{h}=c_{1}y_{1}(t)+c_{2}y_{2}(t)$ +Now we need the particular solution, let $y_{p}$ be in the following form: +$y_{p}(t)=v_{1}(t)y_{1}(t)+v_{2}(t)y_{2}(t)$ <- btw $y_{1}$ and $y_{2}$ are often called a fundamental pair. They are obtained from your homogenous solution. +Impose the following: +1) $v_{1}'y_{1}+v_{2}'y_{2}=0$ +Compute the derivatives and simplify: +$y'_{p}=v_{1}y_{1}'+v_{2}y_{2}'$ +$y_{p}''=v_{1}'y_{1}'+v_{1}y_{1}''+v_{2}'y_{2}'+v_{2}y_{2}''$ +Now we plug those into the second order equation and simplify: +2) $v_{1}'y_{1}'+v_{2}'y_{2}'=\frac{f(t)}{a}$ +We now have a system of two equations (1 and 2). Now we can solve for $v_{1}$ and $v_{2}$: +Using Cramer's rule, we can solve for the system of equations and obtain the solutions: +$v_{1}'=-\frac{{f(t)y_{2}(t)}}{aW[y_{1},y_{2}]}$; $v_{2}'=\frac{{f(t)y_{1}(t)}}{aW[y_{1},y_{2}]}$ <- integrate both sizes to get $v_{1}$ and $v_{2}$. When integrating, you don't need to add a generic constant. +also, $W[y_1,y_{2}]$ is the Wrońskian, and it equals to: $\det \begin{pmatrix}y_{1}&y_{2}\\ y_{1}' &y_{2}'\end{pmatrix}=y_{1}y_{2}'-y_{2}y_{1}'$ +Finally, the general solution is: +$$y(t)=y_{h}+y_{p}\qquad \text{where}\qquad y_{p}(t)=v_{1}(t)y_{1}(t)+v_{2}(t)y_{2}(t)$$ +## What you need to remember +So, what do you need to commit to memory? I believe memorizing these three is a good tradeoff between memory allocated and speed for when you're solving a #voparam problem: +1) $y_{p}(t)=v_{1}(t)y_{1}(t)+v_{2}(t)y_{2}(t)$ +2) $v_{1}'=-\frac{{f(t)y_{2}(t)}}{aW[y_{1},y_{2}]}$ +3) $v_{2}'=\frac{{f(t)y_{1}(t)}}{aW[y_{1},y_{2}]}$ +Alternatively, you could memorize the system of equations and solve for $v_{1}'$ and $v_{2}'$. Ie: +1) $y_{p}(t)=v_{1}(t)y_{1}(t)+v_{2}(t)y_{2}(t)$ +2) $v_{1}'y_{1}+v_{2}'y_{2}=0$ +3) $v_{1}'y_{1}'+v_{2}'y_{2}'=\frac{f(t)}{a}$ +This is what the prof likes. I love you Dr. Minev, but this I personally disagree with. I think I'll stick with the formulas. +--- +#ex #second_order #IVP #voparam #mouc $y''+4y=2\tan(2t)-e^t \qquad y(0)=0 \qquad y'(0)=\frac{4}{5}$ -can we use undetermined coefficients? yes and no -find general solution to homogenous counterpart -1) $y''+4y=0$ -> $r^2+4=0$ -> $r_{1,2}=\pm 2i$ -$y_{h}(t)=c_{1}\cos(2t)+c_{2}\sin(2t)$ -2) $y''+4y=-e^t$ <- use method of undetermined coefficients -$y_{p}^1(t)=Ae^{t}$ +Can we use undetermined coefficients? Yes and no. We can use it on the $e^t$ term. However, guessing and checking $y_{p}$ for the $2\tan(2t)$ term might take a really, really long time. +First, find general solution to homogenous counterpart: +$y''+4y=0$ -> $r^2+4=0$ -> $r_{1,2}=\pm 2i$ +$y_{h}(t)=c_{1}\cos(2t)+c_{2}\sin(2t)$ Done. Easy peasy. +For $-e^t$ lets use method of undetermined coefficients: +$y''+4y=-e^t$ +$y_{p_{1}}(t)=Ae^{t}$ $5Ae^t=-e^t$ $A=-\frac{1}{5}$ -$y_{p}^1(t)=-\frac{1}{5}e^t$ -(ii) $y''+4y=2\tan(2t)$ <- cant use method of undetermined coefficients -$y_{p}^2(t)=v_{1}(t)\cos(2t)+v_{2}(t)\sin(2t)$ -plugging in: -we get a system of eq: - $\cos(2t)v_{1}'+\sin(2t)v_{2}'=0$ - $-2\sin(2t)v_{1}'+2\cos(2t)v_{2}'=2\tan(2t)$ -> we know these two will give a unique solution. ->to solve system of eq multiply each by: ->$2\cos(2t)$ ->$\sin(2t)$ +$y_{p_{1}}(t)=-\frac{1}{5}e^t$ -$2(\sin^2(2t)+\cos^2(2t))v_{2}'=2\tan(2t)\cos(2t)$ -$v_{2}'=\sin(2t)$ -$v_{2}(t)=-\frac{1}{2}\cos(2t)$ no constant of integration, we want one solution only -$v_{1}'=-{\frac{\sin^2(2t)}{\cos(2t)}}$ +Now for $2\tan(2t)$, we cannot realistically use method of undetermined coefficients. +Let's use variation of parameters instead: +$y''+4y=2\tan(2t)$ +$y_{p_{2}}(t)=v_{1}(t)y_{1}(t)+v_{2}(t)y_{2}(t)$ +where $y_{1}=\cos(2t)$, $y_{2}=\sin(2t)$ +recall: +$v_{1}'=-\frac{f(t)y_{2}}{aW[y_{1},y_{2}]}$ +$v_{2}'=\frac{f(t)y_{1}}{aW[y_{1},y_{2}]}$ +plugging in: +$v_{1}'=-\frac{2\tan(2t)\sin(2t)}{(1)(\cos2t(2)\cos2t-\sin2t(-2)\sin2t}=-\frac{2\tan(2t)\sin(2t)}{2\cos^22t+2\sin^22t}=-\tan(2t)\sin(2t)$ +$v_{2}'=\frac{2\tan(2t)\cos(2t)}{2}=\sin(2t)$ <- isn't it nice how we can reuse our computation for the denominator? :D +Now we integrate. +$v_{2}=-\frac{1}{2}\cos(2t)$ <- Don't add a constant of integration, we want one solution only. $v_{1}=-\int \frac{\sin^2(2t)}{\cos(2t)} \, dt$ $v_{1}=-\int \frac{{1-\cos^2(2t)}}{\cos(2t)} \, dt$ $v_1=-\int sec(2t) \, dt+\int \cos(2t) \, dt$ $v_{1}(t)=-\frac{1}{2}\ln\mid sec(2t)+\tan(2t)\mid+\frac{1}{2}\sin(2t)$ -$y_{p}^2(t)=v_{1}(t)\cos(2t)+v_{2}(t)\sin(2t)$ -$y(t)=y_{h}(t)+y_{p}^1(t)+y_{p}^2(t)$ +$y_{p_{2}}(t)=v_{1}(t)\cos(2t)+v_{2}(t)\sin(2t)$ +> ^Now you can start to see how guessing $y_{p_{2}}$ would take a really, really long time. + +$y(t)=y_{h}(t)+y_{p_{1}}(t)+y_{p_{2}}(t)$ =$c_{1}\cos(2t)+c_{2}\sin(2t)+v_{1}(t)\cos(2t)+v_{2}(t)\sin(2t)-\frac{1}{5}e^t$ is the general answer. IVP solution: -$y(0)=0=c_{1}-y_{p}(0)=c_{1}-\frac{1}{5}\Rightarrow c_{1}=\frac{1}{5}$ -skipping some differentiation: $y'(0)=2c_{2}+y_{p}'(0)=2c_{2}+v_{1}'(0)+2v_{2}(0)-\frac{1}{5}=\frac{4}{5}\Rightarrow c_{2}=1$ +$y(0)=0=c_{1}-y_{p}(0)=c_{1}-\frac{1}{5} \implies c_{1}=\frac{1}{5}$ +$y'(0)=\frac{4}{5}=2c_{2}+v_{1}'(0)+2v_{2}(0)-\frac{1}{5} \implies c_{2}=1$ $$y(t)=\frac{1}{5}\cos(2t)+\sin(2t)-\frac{1}{5}e^t+v_{1}(t)\cos(2t)+v_{2}(t)\sin(2t)$$ #end of lecture 9 #start of lecture 10 -# Variation of parameters -last lec we did some variation of parameters -$ay''+by'+cy=f(t)$ -1) $y_{h}(t)=c_{1}y_{1}(t)+c_{2}y_{2}(t)$ -2) $y_{p}(t)=v_{1}(t)y_{1}(t)+v_{2}(t)y_{2}(t)$ - $y_{1}v_{1}'+y_{2}v_{2}'=0$ - $y_{1}v_{1}'+y_{2}'v_{2}'=\frac{b}{a}$ or f/a? -is the system of equations we will need to solve. You can also memorize a formula but peter likes remembering this system of equations and moving on from there. -#ex #variation_of_parameters + +#ex #second_order #voparam #mouc $$y''-2y'+y=e^t\ln(t)+2\cos(t)$$ -i) $y_{h}(t)=?$ +Find homogenous solution first: $r^2-2r+1=0$ -$r_{1,2}=1$ +$r_{1,2}=1$ (repeated root) $y_{h}(t)=c_{1}e^t+c_{2}te^t$ 2) $y_{p}(t)=?$ $y''-2y'+y=2\cos (t)$ -$y_{p}''=A\cos(t)+B\sin(t)$ is our first guess. but it does not solve the homogenous eq. -$y_{p}'=-\sin(t)$ (obtained by using method of undetermined coefficients, computation not shown.) +let's use method of undetermined coefficients: +$y_{p_{1}}=A\cos(t)+B\sin(t)$ is our guess +$y_{p_{1}}'=-A\sin t+B\cos t$ +$y_{p_{1}}''=-A\cos t-B\sin t$ +$-A\cos t-B\sin t+2A\sin t-2B\cos t+A\cos t+B\sin t=2\cos t$ +$-2B\cos t+2A\sin(t)=2\cos t(t)$ +$\implies A=0,\ B=-1$ +$y_{p_{1}}=-\sin(t)$ $y''-2y'+y=e^t\ln(t)$ cant use undetermined coefficients, use variation of parameters $y''_{p}(t)=v_{1}y_{1}+v_{2}y_{2}$ $=v_{1}e^t+v_{2}te^t$ -compute v1 and v2, using the linear system: -eq1) $e^t+v_{1}'+te^tv_{2}'=0$ +Compute v1 and v2. This time lets do it using the linear system for funzies: +eq1) $e^tv_{1}'+te^tv_{2}'=0$ eq2) $e^tv_{1}'+(te^t+e^t){v_{2}'}=e^t{\ln t}$ -subtract eq1 from eq2 $v_{2}'=\ln(t)$ +subtract eq1 from eq2: $v_{2}'=\ln(t)$ $v_{2}(t)=\int \ln(t) \, dt$ integrate by parts $=t\ln(t)-\int t\frac{1}{t} \, dt$ diff --git a/content/_index.md b/content/_index.md index 5eb6829..aaea114 100644 --- a/content/_index.md +++ b/content/_index.md @@ -9,9 +9,12 @@ I have written these notes for myself, I thought it would be cool to share them. [Exact equations (lec 4-5)](exact-equations-lec-4-5.html) [Second order homogenous linear equations (lec 5-7)](second-order-homogenous-linear-equations-lec-5-7.html) [Method of undetermined coefficients (lec 8-9)](method-of-undetermined-coefficients-lec-8-9.html) -[Variation of parameters (lec 9-10)](variation-of-parameters-lec-9-10.html) (raw notes, not reviewed or revised yet.) +[Variation of parameters (lec 9-10)](variation-of-parameters-lec-9-10.html) +[Cauchy-Euler equations (lec 10)](cauchy-euler-equations-lec-10.html) (raw notes, not reviewed or revised yet.) [Free vibrations (lec 11-12)](free-vibrations-lec-11-12.html) (raw notes, not reviewed or revised yet.) -[Resonance in free vibrations (lec 13)](resonance-in-free-vibrations-lec-13.html) (raw notes, not reviewed or revised yet.) +[Resonance in free vibrations (lec 13-14)](resonance-in-free-vibrations-lec-13.html) +[Laplace transform (lec 14)](laplace-transform-lec-14.html) +
[How to solve any DE, a flow chart](Solve-any-DE.png)
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