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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 Sept 20","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":323,"width":466,"height":170},
{"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":"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},
2023-10-01 22:39:08 -06:00
{"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},
2023-09-27 10:58:42 -06:00
{"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},
{"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":"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"}
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