diff --git a/content/Drawing 2023-10-02 13.02.06.excalidraw.md b/content/Drawing 2023-10-02 13.02.06.excalidraw.md new file mode 100644 index 0000000..6c9a7e1 --- /dev/null +++ b/content/Drawing 2023-10-02 13.02.06.excalidraw.md @@ -0,0 +1,34435 @@ +--- + +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": "nN3rWRrjSlk3yv4FhWPvi", + "type": "freedraw", + "x": -404.2604554096621, + "y": -354.5141402360267, + "width": 16.83092320634279, + "height": 373.76338077389994, + "angle": 0, + "strokeColor": "#1e1e1e", + "backgroundColor": "transparent", + "fillStyle": "hachure", + "strokeWidth": 1, + "strokeStyle": "solid", + "roughness": 1, + "opacity": 100, + "groupIds": [], + "frameId": null, + "roundness": 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vibrations (lec 11).md +++ /dev/null @@ -1,56 +0,0 @@ -#start of lecture 11 -last lecture we did cauchy euler equations: -$ax^2{\frac{d^2y}{dx^2}+bx{\frac{ dy }{ dx }}+cy=f(x)},\ x>0$ -where $a,\ b,\ c$ are still constants and $\in \mathbb{R}$ -1) $x=e^t$ -$a{\frac{d^2y}{dt^2}}+(b-a){\frac{dy}{dt}}+cy=f(e^t)$ <- lousy notation, the y here isnt quite the same as in the above definition. -2) $y=x^r$ -$ar^2+(b-a)r+C=0$ -three cases: -(i) $r_1\ne r_{2}$ -then: $y_{h}(x)=c_{1}x^{r_{1}}+c_{2}x^{r_{2}}$ -(ii) $r_{1}=r_{2}=r$ -then: $y_{h}(x)=c_{1}x^r+c_{2}x^r\ln(x)$ -(iii) $r_{1,2}=\alpha+i\beta$ -then: $y_{h}(x)=x^2(c_{1}\cos(\ln \beta x)+c_{2}\sin \ln(\beta x))$ -now find one particular solution for a non homogenous soultion, using variation of parameters, combine the y_h and y_p to get y(x). - -not all equations can fall into cauchy euler type. -$y''+p(x)y'+q(x)y=f(x)$ (1) <- no general solution procudure 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 $y(x)=v(x)y_{1}(x)$ -$y'=v'y_{1}+vy_{1}'$ -$y''=v''y_{1}+2v'y_{1}'+vy_{1}''$ -$(v''y_{1}+2v_{1}'y_{1}'+y_{1}''v)+p(x)(v'y_{1}+vy_{1}')+q(x)vy_{1}=f(x)$ -$v\cancelto{ 0 }{ (y_{1}''+p(x)y_{1}'+q(x)y_{1}) }+v''y_{1}+(2y_{1}'+p(x)y_{1})v'=f$ -$y_{1}v''+()$ -$v''+\left( \frac{2y_{1}'}{y_{1}}+p \right)v'=\frac{f}{y_{1}}$ -$v'=u$ -$u'+\left( \frac{2y_{1}'}{y_{1}}+p \right)u=\frac{f}{y_{1}}$<- 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. - -#ex 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: $y(x)=v(x)y_{1}=v(x)e^{-x^2}$ -$v'=u$ -$u'+\left( \frac{2y_{1}'}{y_{1}}+4x \right)u=\frac{8{e^{-x^2}e^{-2x}}}{e^{-x^2}}$ <-(p(x)=4x) -$u'+\left( \frac{2{e^{-x^2}(-2x)}}{e^{-x^2}}+4x \right)u=8e^{-2x}$ -$u'=8e^{-2x}$ -$u=-4e^{-2x}+c_{1}$ -$v'=u=-4e^{-2x}+c_{1}$ -$v(x)=2e^{-2x}+c_{1}x+c_{2}$ -general solution: -$$y(x)=v(x)y_{1}(x)=(2e^{-2x}+c_{1}x+c_{2})e^{-x^2}$$ - -## Free vibrations -$mr^2+br+k=0$ characteristic polynomail -(i) $r_{1}\ne r_{2}$ $b^2-4mk>0$ -$y_{h}(t)=c_{1}e^{r_{1}t}+c_{2}e^{r_{2}t}$ -$r_{1,2}=-\frac{b}{2m}\pm \frac{\sqrt{ b^2-4mk }}{2m}<0$ -then the limit of the homogenous solution is 0 as t->$\infty$ (over damped case) -(ii) $r_{1}=r_{2}=-\frac{b}{2m}$ -$r_{1}=r_{2}=-\frac{b}{2m}$ -$y_{h}(t)=e^-\frac{b}{2m}+c_{2}te^{-b/2m}t$ limit =0 as t approches inf critically damped \ No newline at end of file diff --git a/content/Free vibrations (lec 11-12).md b/content/Free vibrations (lec 11-12).md new file mode 100644 index 0000000..4f3f67e --- /dev/null +++ b/content/Free vibrations (lec 11-12).md @@ -0,0 +1,120 @@ +#start of lecture 11 +last lecture we did cauchy euler equations: +$ax^2{\frac{d^2y}{dx^2}+bx{\frac{ dy }{ dx }}+cy=f(x)},\ x>0$ +where $a,\ b,\ c$ are still constants and $\in \mathbb{R}$ +1) $x=e^t$ +$a{\frac{d^2y}{dt^2}}+(b-a){\frac{dy}{dt}}+cy=f(e^t)$ <- lousy notation, the y here isnt quite the same as in the above definition. +2) $y=x^r$ +$ar^2+(b-a)r+C=0$ +three cases: +(i) $r_1\ne r_{2}$ +then: $y_{h}(x)=c_{1}x^{r_{1}}+c_{2}x^{r_{2}}$ +(ii) $r_{1}=r_{2}=r$ +then: $y_{h}(x)=c_{1}x^r+c_{2}x^r\ln(x)$ +(iii) $r_{1,2}=\alpha+i\beta$ +then: $y_{h}(x)=x^2(c_{1}\cos(\ln \beta x)+c_{2}\sin \ln(\beta x))$ +now find one particular solution for a non homogenous soultion, using variation of parameters, combine the y_h and y_p to get y(x). + +not all equations can fall into cauchy euler type. +$y''+p(x)y'+q(x)y=f(x)$ (1) <- no general solution procudure 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 $y(x)=v(x)y_{1}(x)$ +$y'=v'y_{1}+vy_{1}'$ +$y''=v''y_{1}+2v'y_{1}'+vy_{1}''$ +$(v''y_{1}+2v_{1}'y_{1}'+y_{1}''v)+p(x)(v'y_{1}+vy_{1}')+q(x)vy_{1}=f(x)$ +$v\cancelto{ 0 }{ (y_{1}''+p(x)y_{1}'+q(x)y_{1}) }+v''y_{1}+(2y_{1}'+p(x)y_{1})v'=f$ +$y_{1}v''+()$ +$v''+\left( \frac{2y_{1}'}{y_{1}}+p \right)v'=\frac{f}{y_{1}}$ +$v'=u$ +$u'+\left( \frac{2y_{1}'}{y_{1}}+p \right)u=\frac{f}{y_{1}}$<- 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. + +#ex 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: $y(x)=v(x)y_{1}=v(x)e^{-x^2}$ +$v'=u$ +$u'+\left( \frac{2y_{1}'}{y_{1}}+4x \right)u=\frac{8{e^{-x^2}e^{-2x}}}{e^{-x^2}}$ <-(p(x)=4x) +$u'+\left( \frac{2{e^{-x^2}(-2x)}}{e^{-x^2}}+4x \right)u=8e^{-2x}$ +$u'=8e^{-2x}$ +$u=-4e^{-2x}+c_{1}$ +$v'=u=-4e^{-2x}+c_{1}$ +$v(x)=2e^{-2x}+c_{1}x+c_{2}$ +general solution: +$$y(x)=v(x)y_{1}(x)=(2e^{-2x}+c_{1}x+c_{2})e^{-x^2}$$ + +## Free vibrations +$mr^2+br+k=0$ characteristic polynomial +(i) $r_{1}\ne r_{2}$ $b^2-4mk>0$ +$y_{h}(t)=c_{1}e^{r_{1}t}+c_{2}e^{r_{2}t}$ +$r_{1,2}=-\frac{b}{2m}\pm \frac{\sqrt{ b^2-4mk }}{2m}<0$ +then the limit of the homogenous solution is 0 as t->$\infty$ (over damped case) +(ii) $r_{1}=r_{2}=-\frac{b}{2m}$ +$r_{1}=r_{2}=-\frac{b}{2m}$ +$y_{h}(t)=e^-\frac{b}{2m}+c_{2}te^{-b/2m}t$ limit =0 as t approaches inf (critically damped) + +#end of lec 11 #start of lec 12 (oct 2 2023) +![[Drawing 2023-10-02 13.02.06.excalidraw]] +let $\omega =\frac{\sqrt{ 4mk-b^2 }}{2m}$ (angular frequency) +then the underdamped case is: +$y(t)=(c_{1}\cos \omega t+c_{2}\sin \omega t)e^{\frac{-b}{2m}t}$ +we know the trig identity: +$\sin(\alpha+\beta)=\sin \alpha\cos \beta+\cos \alpha \sin \beta$ +cant make c_1 c_2 sin or cos so what we do? +do a power transform to convert cartesian into cylindrical coordinates +$c_{1}=A\sin \phi$ +$c_{2}=A\cos \phi$ +then: +$Ae^{-bt/2m}(\sin \phi \cos \omega t+\cos \phi \sin \omega t)$ +$=Ae^{-bt/2m}\sin(\omega t+\phi)$ where $\phi$ is the phase shift. +and $\frac{\omega}{2\pi}$ is the natural frequency +$\frac{2\pi}{\omega}$ is the period +but this is all classical mechanics, but beatifully the world of electronic circuits of R L C also has these equations. Biology too. Nature is beautiful and harmonic. +btw we know $A=\sqrt{ c_{1}^2+c_{2}^2 }$ +and $\tan \phi=\frac{c_{1}}{c_{2}}$ +so we can get A and phi from c_1 and c_2. +this under damped case also reaches 0 as t->$\infty$ + +this system in the drawing is in free vibrattion (RHS=0 means no external force=free vibration.) +#ex +$y''+by'+25y=0 \qquad y(0)=1\quad y'(0)=0$ +1) b=0 -> no friction in the system (undamped) +$b^2-4mk$ +$y(t)=c_{1}\cos 5t+c_{2}\sin 5t$ +$y(0)=c_1=1$ +$y'(0)=0=c_{2}$ +then $\sin 5t\Rightarrow y(t)=\cos(5t)=\sin\left( 5t+\frac{\pi}{2} \right)$ (by trig identity) +important take away from undamped case: amplitude is constant 1, oscillates forever. +2) b=6 +compute $b^2-4mk=36-4*25=-64$ +$r_{1,2}=-\frac{6}{2}\pm4i$ +$y(t)=e^{-3t}(c_{1}\cos4t+c_{2}\sin4t)$ +still under damped situation. +$y(0)=1=c_{1}$ +$y'(0)=0=-3c_{1}+4c_{2}\Rightarrow c_{2}=\frac{3}{4}$ +$A=\frac{5}{4}$ +$\tan \phi=\frac{4}{3}$ +$\phi \approx 0.9273\dots$ +$$y(t)=\frac{5}{4}e^{-3t}\sin(4t+\phi)$$ +"I know engineers loves calculators, I know mathematicians hates calculators, and that's probably the only difference between mathematicians and engineers." -Peter (referring to calculating arctan(4/3) on an exam) +3) b=10 +$r_{1,2}=-5$ +$y(t)=(c_{1}+c_{2}t)e^{-5t}$ +$y(0)=1=c_{1}$ +$y'(0)=c_{2}-5c_{1}=0$ +$c_{2}=5$ +$y(t)=(1+5t)e^{-5t}\rightarrow0_{as\ t\to\infty}$ +$y(t)=(1+5t)e^{-5t}>0$ +4) b=12 + $r_{1,2}=-6\pm \sqrt{ 11 }$ +$y(t)=c_{1}e^{(-6\pm \sqrt{ 11 })t}+c_{2}e^{(-6-\sqrt{11 })t}$ +$y(0)=c_{1}+c_{2}=1$ +$y'(0)=(-6+\sqrt{ 11 })c_{1}+(-6-\sqrt{ 11 })c_{2}=0$ +$c_{1}=\frac{11+6\sqrt{ 11 }}{22}$ +$c_{2}=\frac{{11-6\sqrt{ 11 }}}{22}$ +this is an over damped case. + +lets look at the graphs: (graphs featuring the three cases shown on projector.) +#end of lec 12 \ 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 1f293f2..33c2d80 100644 --- a/content/Method of undetermined coefficients (lec 8-9).md +++ b/content/Method of undetermined coefficients (lec 8-9).md @@ -11,7 +11,7 @@ if $r_{1}=r_{2}\Rightarrow y_{h}(t)=c_{1}e^{rt}+c_{2}te^{rt}$ <- **critically da if $r_{1,2}\in \mathbb{C}\Rightarrow y_{h}(t)=e^{\alpha t}(c_{1}\cos(\beta t)+c_{2}\sin(\beta t))$ <- **underdamped** where h means homogenous, (when $f(t)=0$ the equation is homogenous.) -in the case when $f(t)\ne 0$ : +But what about the case when $f(t)\ne 0$ ? 2) If $y_{p}(t)$ solves 1) then the general solution to $y(t)$ is $y(t)=y_{h}(t)+y_{p}(t)$ theorem: if $p(t),\ q(t),\ f(t)$ are continuous on $I$ then the following IVP has a unique solution: $y''+p(t)y'+q(t)y=f(t)\quad \text{where}\quad y''(t_{o}),\ y'(t_{o}),\ y(t_{o})\in I$ --- @@ -106,4 +106,4 @@ case ii) $ay''+by'+cy=P_{m}(t)e^{\alpha t}\cos(\beta t)+P_{m}(t)e^{\alpha t}\sin 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: 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 +s=1 if $\alpha+i\beta$ is a root. \ No newline at end of file diff --git a/content/Solve-any-DE.canvas b/content/Solve-any-DE.canvas index 225d904..1d67852 100644 --- a/content/Solve-any-DE.canvas +++ b/content/Solve-any-DE.canvas @@ -3,7 +3,7 @@ {"id":"767241b95828457e","type":"text","text":"Integrate both sides","x":-57,"y":-360,"width":352,"height":80}, {"id":"3572d3ccf3a666dc","type":"text","text":"# Solution","x":-43,"y":-520,"width":325,"height":90}, {"id":"ac4eb08e6ceeccbd","type":"text","text":"# How to solve (almost) any differential equation:","x":-561,"y":-760,"width":1296,"height":136}, - {"id":"76786ab85409e204","type":"text","text":"updated Sept 20","x":778,"y":-729,"width":301,"height":74}, + {"id":"76786ab85409e204","type":"text","text":"updated Oct 1","x":778,"y":-729,"width":301,"height":74}, {"id":"c6a536ee57248cfd","type":"file","file":"Math 201/Lectures/Separable equations (lec 1).md","x":-102,"y":-170,"width":441,"height":140}, {"id":"df03f8f1000d374a","type":"text","text":"If $y'=G(ax+by)$ \n\nsubstitute u=ax+by\n$\\frac{du}{dx}=a+b\\frac{ dy }{ dx}$","x":-251,"y":121,"width":341,"height":165}, {"id":"1044e4c2c0610a1e","type":"text","text":"if $y'=G\\left( \\frac{y}{x} \\right)$ \n\nsubstitute $u=\\frac{y}{x}$\n$\\frac{dy}{dx}=u+x{\\frac{du}{dx}}$","x":150,"y":121,"width":315,"height":165}, @@ -20,11 +20,6 @@ {"id":"ece672db8e16ac5a","type":"text","text":"substitute $x=u+k$\n$y=v+l$","x":9,"y":769,"width":242,"height":156}, {"id":"120b3008bd08d69a","type":"file","file":"Math 201/Lectures/Linear coefficient equations (lec 4).md","x":-81,"y":1035,"width":422,"height":143}, {"id":"bebd67e847df16e1","type":"text","text":"Shortcut:\n$$I(x)=e^{\\int (1-n)P(x) \\, dx }$$\n$$y^{1-n}=\\frac{1}{I(x)}\\left( \\int (1-n)I(x)Q(x) \\, dx +C\\right)$$","x":-1333,"y":-430,"width":505,"height":260}, - {"id":"3f081acda4f30a27","type":"text","text":"solve for $r_{1}$ & $r_{2}$ using quadratic formula","x":1315,"y":183,"width":274,"height":125}, - {"id":"4ffaa5c9a7e8d22b","type":"text","text":"use principle of super position","x":1329,"y":29,"width":245,"height":96}, - {"id":"cd7490f8cce0b6e0","type":"file","file":"Math 201/Lectures/Second order homogenous linear equations (lec 5-7).md","x":1215,"y":780,"width":474,"height":145}, - {"id":"cd31ca74652b6936","type":"text","text":"substitute $y(t)=e^{rt}$ and its derivatives in the equation","x":1293,"y":556,"width":317,"height":122}, - {"id":"e063ab92aef817e4","type":"text","text":"divide both sides of equation by $e^{rt}$","x":1322,"y":382,"width":260,"height":110}, {"id":"d98da52cb7139c25","type":"text","text":"$N(x,y)=\\frac{\\partial}{\\partial y} \\int M \\, dx+g(y)$\n","x":514,"y":354,"width":328,"height":77}, {"id":"ba1ef733f104894d","type":"text","text":"$M(x,y)=\\frac{\\partial}{\\partial x} \\int N \\, dy+g(x)$\n","x":875,"y":354,"width":340,"height":77}, {"id":"3cc6d2966364d19c","type":"text","text":"$F(x,y)=\\int M \\, dx+g(y)$","x":520,"y":497,"width":315,"height":94}, @@ -35,9 +30,28 @@ {"id":"d172c6ad7421b458","type":"file","file":"Math 201/Lectures/Exact equations (lec 4-5).md","x":674,"y":903,"width":393,"height":133}, {"id":"d3ffeeccd9e88489","type":"text","text":"$M(x,y)dx+N(x,y)dy=0$","x":674,"y":1036,"width":342,"height":83}, {"id":"937768cc91c15daa","type":"text","text":"$(a_{1}x+b_{1}y+c_{1})dx+(a_{2}x+b_{2}y+c_{2})dy=0$","x":-81,"y":1178,"width":516,"height":63}, - {"id":"59f62d39b48e7b57","x":1663,"y":-306,"width":250,"height":60,"type":"text","text":"under-damped"}, - {"id":"7089887c01722e83","x":1880,"y":-173,"width":250,"height":60,"type":"text","text":"over-damped"}, - {"id":"b3e73030feee12da","x":1843,"y":33,"width":287,"height":50,"type":"text","text":"critically-damped"} + {"id":"b3e73030feee12da","type":"text","text":"case 2)\ncritically-damped\n$r_{1}=r_{2}=r$","x":1420,"y":177,"width":300,"height":140}, + {"id":"59f62d39b48e7b57","type":"text","text":"case 3)\nunder-damped\n$r_{1,2}=\\alpha\\pm \\beta i$","x":1760,"y":177,"width":280,"height":140}, + {"id":"7089887c01722e83","type":"text","text":"case 1)\nover-damped $r_{1}\\ne r_{2}$","x":1140,"y":177,"width":240,"height":140}, + {"id":"fb1b8ea1fa7e997b","type":"text","text":"general solution:\n$y_{h}(t)=c_{1}e^{r_{1}t}+c_{2}e^{r_{2}t}$","x":1079,"y":-83,"width":281,"height":103}, + {"id":"21dddcaf48c1f19e","type":"text","text":"general solution:\n$y_{h}(t)=c_{1}e^{rt}+c_{2}te^{rt}$","x":1410,"y":-83,"width":320,"height":90}, + {"id":"e85838ac18ea4a1e","type":"text","text":"general solution:\n$y_{h}(t)=e^{\\alpha t}(c_{1}\\cos\\beta t+c_{2}\\sin\\beta t)$","x":1760,"y":-83,"width":380,"height":103}, + {"id":"cd7490f8cce0b6e0","type":"file","file":"Math 201/Lectures/Second order homogenous linear equations (lec 5-7).md","x":1336,"y":808,"width":465,"height":190}, + {"id":"e063ab92aef817e4","type":"text","text":"obtain characteristic equation","x":1420,"y":619,"width":298,"height":78}, + {"id":"d9167b6da862a39f","type":"file","file":"Math 201/Lectures/Free vibrations (lec 11-12).md","x":1467,"y":-875,"width":433,"height":146}, + {"id":"9d6ed2c739b0615e","type":"file","file":"Math 201/Lectures/Variation of parameters (lec 9-10).md","x":1467,"y":-655,"width":530,"height":153}, + {"id":"2b3a2c1a51539afd","type":"text","text":"idk how to solve these yet","x":1139,"y":-802,"width":241,"height":102}, + {"id":"3f081acda4f30a27","type":"text","text":"solve for $r_{1}$ & $r_{2}$ using quadratic formula $\\frac{{-b\\pm \\sqrt{ b^2-4ac }}}{2a}$","x":1365,"y":419,"width":408,"height":125}, + {"id":"50996e52b63ee5e0","type":"text","text":"$ay''+by'+cy=0$","x":1336,"y":998,"width":250,"height":60}, + {"id":"174f52023f715bf2","type":"file","file":"Math 201/Lectures/Method of undetermined coefficients (lec 8-9).md","x":2240,"y":1143,"width":498,"height":196}, + {"id":"0daa567ae142c34f","type":"text","text":"split into the homogenous solution and particular solution","x":2313,"y":927,"width":352,"height":131}, + {"id":"e5369edab4d8a607","type":"text","text":"find $y_{h}(t)$\n(the easy part)","x":1911,"y":895,"width":229,"height":98}, + {"id":"208cafba1dc71798","type":"text","text":"Use intuition and guess and check. (Practice improves intuition)","x":2068,"y":247,"width":332,"height":132}, + {"id":"eded7f0b26c3ebc5","type":"text","text":"use generalized algorithm:\ncase 1) $ay''+by'+cy=P_{m}(t)e^{rt}$\nthen: $y_{p}(t)=t^s(b_{m}t^m+b_{m-1}t^{m-1}+\\dots+b_{0})e^{rt}$\nwhere:\ns=0, if r is not a root,\ns=1 if r is a single root,\ns=2 if r is a double root\n\ncase 2)\n$ay''+by'+cy=P_{m}(t)e^{\\alpha t}\\cos(\\beta t)+P_{m}(t)e^{\\alpha t}\\sin(\\beta t)$\nthen: $y_{p}(t)=t^s[(A_{k}t^k+A_{K-1}t^{k-1}+\\dots+A_{0})e^{\\alpha t}\\cos(\\beta t)+(B_{k}t^k+B_{k-1}t^{k-1}+\\dots+B_{0})e^{\\alpha t}\\sin(\\beta t)]$\nwhere:\ns=0 if $\\alpha+i\\beta$ is not a root,\ns=1 if $\\alpha+i\\beta$ is a root.","x":2489,"y":-156,"width":1054,"height":575}, + {"id":"d327b2b2e59fc4cc","type":"text","text":"general solution will be $y(t)=y_{h}(t)+y_{p}(t)$","x":2108,"y":-261,"width":336,"height":105}, + {"id":"b9455e68aa57c635","type":"text","text":"$ay''+by'+cy=f(t)$","x":2239,"y":1339,"width":283,"height":59}, + {"id":"5ef326cd80c96f9a","type":"text","text":"if $f(t)$ is a sum of two functions, split it up and find $y_{p_{1}}$, $y_{p_{2}}, \\dots$ separately, then $y_{p}$ will be the sum of the all the $y_{p_{n}}$'s you find.","x":2276,"y":658,"width":426,"height":188}, + {"id":"ac0242af88518588","type":"text","text":"find $y_{p}(t)$ (hard part)","x":2389,"y":520,"width":200,"height":99} ], "edges":[ {"id":"7f7a9f03d9ef74ef","fromNode":"c6a536ee57248cfd","fromSide":"top","toNode":"767241b95828457e","toSide":"bottom"}, @@ -55,13 +69,10 @@ {"id":"bbc151d3f261f179","fromNode":"5263b3c4f5b28ec6","fromSide":"top","toNode":"3572d3ccf3a666dc","toSide":"left"}, {"id":"473fd48634427ca8","fromNode":"a662aed47cf28581","fromSide":"top","toNode":"ba1ef733f104894d","toSide":"bottom"}, {"id":"c3fd9e04ae24009a","fromNode":"3cc6d2966364d19c","fromSide":"top","toNode":"d98da52cb7139c25","toSide":"bottom"}, - 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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: $a(v_{1}'y_{1}'+v_{1}y_{1}''+v_{2}'y_{2}'+v_{2}y_{2}'')+b(v_{1}y_{1}'+v_{2}y_{2}')+c(v_{1}y_{1}+v_{2}y_{2})=f(t)$ -$v_{1}(ay_{1}''+\cancelto{ 0 }{ by_{1}' }+cy_{1})+v_{2}(ay_{2}''+\cancelto{ 0 }{ by_{2}'C }+cy_{2})+a(v_{1}'y_{1}'+v_{2}'y_{2}')$ +$v_{1}(\cancel{ ay_{1}''+ by_{1}' +cy_{1} })+v_{2}(\cancel{ ay_{2}''+ by_{2}'+cy_{2} })+a(v_{1}'y_{1}'+v_{2}'y_{2}')$ +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 v1,2. When integrating, you don't need to add a generic constant. +$v_{1}'=\frac{{f(t)y_{2}t}}{aW[y_{1},y_{2}]}$; $v_{2}'=-\frac{{f(t)y_{1}(t)}}{aW[y_{1},y_{2}]}$ <- integrate both sizes to get $v_{1}$ and $v_{2}$. When integrating, you don't need to add a generic constant. #ex #second_order #IVP $y''+4y=2\tan(2t)-e^t \qquad y(0)=0 \qquad y'(0)=\frac{4}{5}$ @@ -21,11 +24,11 @@ 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 -$y_{p}'(t)=Ae^{t}$ +2) $y''+4y=-e^t$ <- use method of undetermined coefficients +$y_{p}^1(t)=Ae^{t}$ $5Ae^t=-e^t$ $A=-\frac{1}{5}$ -$y_{p}'(t)=-\frac{1}{5}e^t$ +$y_{p}^1(t)=-\frac{1}{5}e^t$ (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: @@ -41,19 +44,19 @@ $2(\sin^2(2t)+\cos^2(2t))v_{2}'=2\tan(2t)\cos(2t)$ $v_{2}'=\sin(2t)$ $v_{2}(t)=-\frac{1}{2}\cos(2t)$ no constant of integration, we want one solution only $v_{1}'=-{\frac{\sin^2(2t)}{\cos(2t)}}$ -$v_{1}=-\int \frac{\sin^2(2t)}{\cos(2t)} \, dx$ +$v_{1}=-\int \frac{\sin^2(2t)}{\cos(2t)} \, dt$ $v_{1}=-\int \frac{{1-\cos^2(2t)}}{\cos(2t)} \, dt$ -$v_1=-\int sec(2t) \, dx+\int \cos(2t) \, dt$ +$v_1=-\int sec(2t) \, dt+\int \cos(2t) \, dt$ $v_{1}(t)=-\frac{1}{2}\ln\mid sec(2t)+\tan(2t)\mid+\frac{1}{2}\sin(2t)$ $y_{p}^2(t)=v_{1}(t)\cos(2t)+v_{2}(t)\sin(2t)$ -$y(t)=y_{h}(t)+y_{p}'(t)+y_{p}^2(t)$ +$y(t)=y_{h}(t)+y_{p}^1(t)+y_{p}^2(t)$ =$c_{1}\cos(2t)+c_{2}\sin(2t)+v_{1}(t)\cos(2t)+v_{2}(t)\sin(2t)-\frac{1}{5}e^t$ -is our general answer. +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: $y'(0)=2c_{2}+y_{p}'(0)=2c_{2}+v_{1}'(0)+2v_{2}(0)-\frac{1}{5}=\frac{4}{5}\Rightarrow c_{2}=1$ -$y(t)=\frac{1}{5}\cos(2t)+\sin(2t)-\frac{1}{5}e^t+v_{1}(t)\cos(2t)+v_{2}(t)\sin(2t)$ +$$y(t)=\frac{1}{5}\cos(2t)+\sin(2t)-\frac{1}{5}e^t+v_{1}(t)\cos(2t)+v_{2}(t)\sin(2t)$$ #end of lecture 9 #start of lecture 10 diff --git a/content/_index.md b/content/_index.md index 8c670a9..5be0f6f 100644 --- a/content/_index.md +++ b/content/_index.md @@ -10,7 +10,7 @@ I have written these notes for myself, I thought it would be cool to share them. [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.) -[Free vibrations (lec 11)](free-vibrations-lec-11.html) (raw notes, not reviewed or revised yet.) +[Free vibrations (lec 11-12)](free-vibrations-lec-11-12) (raw notes, not reviewed or revised yet.)
[How to solve any DE, a flow chart](Solve-any-DE.png)
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