MATH201/content/Solve-any-DE.canvas

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		{"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 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},
		{"id":"c280d80abc2ea256","type":"file","file":"Math 201/Lectures/Homogenous equations (lec 2).md","x":-91,"y":380,"width":441,"height":149},
		{"id":"ffbe5ed5493f9419","type":"text","text":"find $\\mu(x)$","x":-561,"y":156,"width":250,"height":60},
		{"id":"78362e72fb0d54af","type":"file","file":"Math 201/Lectures/Linear equations (lec 2-3).md","x":-777,"y":354,"width":466,"height":139},
		{"id":"e71a5b824ac543f8","type":"text","text":"combine terms using product rule","x":-590,"y":-30,"width":309,"height":103},
		{"id":"70316d4131dc52c3","type":"file","file":"Math 201/Lectures/Bernoulli equations (lec 3).md","x":-884,"y":973,"width":465,"height":136},
		{"id":"e70d1e3eea85e227","type":"text","text":"substitute","x":-669,"y":601,"width":250,"height":60},
		{"id":"ba7ff8f24112635b","type":"text","text":"$\\frac{d}{dx}(y^{1-n}=u)$","x":-669,"y":709,"width":250,"height":60},
		{"id":"f42f6dfefb957902","type":"text","text":"let $y^{1-n}=u$","x":-669,"y":811,"width":250,"height":60},
		{"id":"5263b3c4f5b28ec6","type":"text","text":"Shortcut: $$y(x)=\\frac{1}{I(x)}\\int I(x)Q(x) \\, dx $$","x":-1006,"y":-37,"width":355,"height":193},
		{"id":"109d20a863d93116","type":"text","text":"Solve system of linear equations","x":-28,"y":601,"width":316,"height":96},
		{"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":"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},
		{"id":"a662aed47cf28581","type":"text","text":"$F(x,y)=\\int N \\, dy+g(x)$","x":894,"y":497,"width":302,"height":94},
		{"id":"5dc72ba2fdd9b5af","type":"text","text":"solve for $g$","x":735,"y":177,"width":250,"height":60},
		{"id":"dc3f64df05e215d4","type":"text","text":"use $F(x,y)=C$ and the $F(x,y)$ you got from the integral to obtain the general solution","x":665,"y":-83,"width":390,"height":180},
		{"id":"90a401b5afaf25e6","type":"text","text":"Test for exactness:\n$\\frac{\\partial M}{\\partial y}=\\frac{\\partial N}{\\partial x}$","x":732,"y":697,"width":277,"height":102},
		{"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":"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 i) $ay''+by'+cy=P_m(t)e^{rt}$\nwhere $P_{m}(t)=a_{m}t^m+a_{m-1}t^{m-1}+ \\dots +a_{0}$ i.e., $P$ is a polynomial degree $m$.\nThen we guess the particular solution is of the form: $y_{p}(t)=t^s(b_{m}t^m+b_{m-1}t^{m-1}+\\dots+b_{0})e^{rt}$\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 ii) $ay''+by'+cy=P_{m}(t)e^{\\alpha t}\\cos(\\beta t)+Q_{n}(t)e^{\\alpha t}\\sin(\\beta t)$\nThen we guess the particular solution is of the form: $y_{p}(t)=t^s[(A_{k}t^k+A_{k-1}t^{k-1}+\\dots+A_{0})e^{\\alpha t}\\cos(\\beta t)+(B_{k}t^k+B_{k-1}t^{k-1}+\\dots+B_{0})e^{\\alpha t}\\sin(\\beta t)]$\nwhere:\n$k=\\max(m,n)$\n$s=0$ if $\\alpha+i\\beta$ is not a root,\n$s=1$ if $\\alpha+i\\beta$ is a root.  ","x":2489,"y":-156,"width":1054,"height":596},
		{"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}
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