# Blackboard Shots with Prefix "12_267"

12_267 is 2012 MAT 267 - Advanced ODEs.

121204-100949: The amplitudes when $q\to L$ (3).

121204-100307: The amplitudes when $q\to L$ (2).

121204-095516: The amplitudes when $q\to L$.

121204-095148: Bounding amplitudes on the other side.

121204-094026: The basic amplitudes theorem (2).

121204-093507: The basic amplitudes theorem.

121204-093124: Chaninging the independent variable (2).

121204-092337: Chaninging the independent variable.

121204-092329: Notes.

121203-100051: Changing the independent variable (2).

121203-095636: Changing the independent variable.

121203-094744: The Sturm comparison theorem - comparing with Euler (3).

121203-094157: The Sturm comparison theorem - comparing with Euler (2).

121203-093704: The Sturm comparison theorem - comparing with Euler.

121203-093037: The Sturm comparison theorem - studying Bessel.

121203-092424: The Sturm comparison theorem - self comparisons (2).

121203-091617: The Sturm comparison theorem - self comparisons.

121203-091524: Notes and riddle.

121130-100324: More on $y''+x^\alpha y=0$.

121130-095719: The Sturm Comparison Theorem (3).

121130-095336: The Sturm Comparison Theorem (2).

121130-095035: The Sturm Comparison Theorem.

121130-094519: $y''+x^\alpha y=0$.

121130-094107: The non-oscillation theorem (5).

121130-093715: The non-oscillation theorem (4).

121130-093107: The non-oscillation theorem (3).

121130-092351: The non-oscillation theorem (2).

121130-091916: The non-oscillation theorem.

121130-091418: Reminders

121130-091002: Today's Catalan.

121127-100434: Changing the dependent variable (3).

121127-095841: Changing the dependent variable (2).

121127-095243: Changing the dependent variable.

121127-094605: The basic oscillation theorem (3).

121127-094055: The basic oscillation theorem (2).

121127-093543: The basic oscillation theorem.

121127-093006: Restoring forces, the case of $q<0$.

121127-091026: Airy's equation - why?

121127-091017: Announcements.

121126-100200: The hardest case - $\alpha_1-\alpha_2\in{\mathbb N}_{>0}$.

121126-095126: The case of a double root (2).

121126-094748: The case of a double root.

121126-094123: Faking the graph of $x^{1/2}\cos(\frac12\log x)$.

121126-093828: The easy case with complex numbers.

121126-093307: The easy case.

121126-092606: The fundamental series of $J_{1/3}$.

121126-091701: Reminders, the fundamental series.

121123-100409: RSP at order 2 (5).

121123-100024: RSP at order 2 (4).

121123-095637: RSP at order 2 (3).

121123-095031: RSP at order 2 (2).

121123-094756: RSP at order 2.

121123-094145: RSP at order 1 (4).

121123-093629: RSP at order 1 (3).

121123-093057: RSP at order 1 (2).

121123-092500: RSP at order 1.

121123-091716: Today's topics.

121123-091132: Riddle along.

121120-100237: Proof of Fuchs' theorem (3).

121120-095532: Proof of Fuchs' theorem (2).

121120-094617: Proof of Fuchs' theorem.

121120-093509: Fuchs' theorem.

121120-092913: The Airy equation by power series (4).

121120-091425: The Airy equation by power series (3).

121119-100238: The Airy equation by power series (2).

121119-095656: The Airy equation by power series.

121119-095030: Examples for functions given by a formula (2).

121119-094324: Examples for functions given by a formula.

121119-093656: On functions given by a formula.

121119-093305: The radius of convergence of a series (2).

121119-092529: The radius of convergence of a series.

121119-092128: $\pi$ is irrational.

121119-091004: A proposition by Samer Seraj.

121116-100323: A bit about convergence of series (2).

121116-095425: A bit about convergence of series.

121116-094936: Solving using power series (2).

121116-094011: Solving using power series.

121116-092615: Power series - motivation.

121116-091938: Wronskians and $\cos^2 x + \sin^2 x$.

121116-091126: Riddle Along.

121109-100032: The case of 2nd order linear ODEs.

121109-095425: Differetiating the Wronskian.

121109-094907: Differetiating derivatives (3).

121109-094707: Differetiating derivatives (2).

121109-094123: Differetiating derivatives.

121109-093513: The Wronskian.

121109-093219: Global existence for linear systems (2).

121109-092640: Global existence for linear systems.

121109-091529: Claims and Debts of systems of ODEs.

121106-215845: Challenges.

121106-215829: A differential equation for the generating function of the $A_n$ (2).

121106-215243: A differential equation for the generating function of the $A_n$.

121106-215031: A recursion for $A_n$.

121106-214712: The generating function of $C_n$ (2).

121106-214107: The generating function of $C_n$.

121106-213547: A recursive formula for the Catalan numbers $C_n$.

121106-212459: $A_n$ and $C_n$.

121106-212016: Debts on systems.

121106-210901: Riddles Along.

121105-110135: Proof of the invertibility claim (2).

121105-105750: Proof of the invertibility claim.

121105-105456: The non-homogeneous case using a Fundamental Matrix (4).

121105-105104: The non-homogeneous case using a Fundamental Matrix (3).

121105-104645: The non-homogeneous case using a Fundamental Matrix (2).

121105-104144: The non-homogeneous case using a Fundamental Matrix.

121105-103615: The non-homogeneous case by diagonalization (5).

121105-103301: The non-homogeneous case by diagonalization (4).

121105-102738: The non-homogeneous case by diagonalization (3).

121105-102024: The non-homogeneous case by diagonalization (2).

121105-101538: The non-homogeneous case by diagonalization.

121030-095809: Example with a repeated eigenvalue (2).

121030-095415: Example with a repeated eigenvalue.

121030-095102: Exponentiating a Jordan block.

121030-094321: The Jordan form theorem (2).

121030-093500: The Jordan form theorem.

121030-093236: Example with distinct eigenvalues (3).

121030-092306: Example with distinct eigenvalues (2).

121030-091915: Example with distinct eigenvalues.

121030-091559: Reminders.

121030-091036: Announcements.

121029-100048: Properties of matrix exponentiation (6).

121029-095744: Properties of matrix exponentiation (5).

121029-095326: Properties of matrix exponentiation (4).

121029-094547: Properties of matrix exponentiation (3).

121029-094025: Properties of matrix exponentiation (2).

121029-093558: Properties of matrix exponentiation.

121029-092858: Convergence.

121029-091928: Exponentiation via the Taylor series.

121029-091107: Announcements.

121023-100040: Matrix exponentiation (2).

121023-095938: Matrix exponentiation.

121023-095149: A baby version.

121023-094809: Systems of linear equations.

121023-093831: Undetermined coefficients (4).

121023-092351: Undetermined coefficients (3).

121023-091741: Undetermined coefficients (2).

121023-091718: Pre-exam office hours.

121022-100052: Undetermined coefficients.

121022-095206: Reduction of order.

121022-094433: Multiple roots (5).

121022-093955: Multiple roots (4).

121022-093804: Multiple roots (3).

121022-093504: An aside on the Leibniz rule for higher derivatives.

121022-092921: Multiple roots (2).

121022-092104: Multiple roots.

121022-091332: The case of distinct roots.

121022-090631: TT, Read Along, Riddle Along.

121019-095816: From complex back to real.

121019-095806: Distinct real roots, complex root.

121019-094930: Differential operator language.

121019-094408: The guessing method.

121019-094033: Constant coefficients homogeneous high order ODEs (2).

121019-094024: Constant coefficients homogeneous high order ODEs.

121019-093128: Numerical Integration (3).

121019-093058: Numerical Integration (2).

121019-091840: Numerical Integration.

121016-093936: Runge-Kutta.

121016-093634: A general scheme.

121016-093057: Local analysis of improved Euler (2).

121016-092401: Local analysis of improved Euler.

121016-091448: Euler and improved Euler.

121016-090737: Term test info and riddle.

121012-095603: Numerical methods, starting from the silly (3).

121012-095555: Numerical methods, starting from the silly (2).

121012-094800: Numerical methods, starting from the silly.

121012-093338: Lagrange multipliers in CoV.

121012-092651: The Lagrange Multipliers Theorem (3).

121012-092050: The Lagrange Multipliers Theorem (2).

121012-091426: The Lagrange Multipliers Theorem.

121012-090625: Read along and riddle along.

121009-095640: Directional derivatives.

121009-095113: The isoperimetric inequality (4).

121009-094819: The isoperimetric inequality (3).

121009-094131: The isoperimetric inequality (2).

121009-093631: Lagrange multipliers in ${\mathbb R}^2$ (4).

121009-093049: Lagrange multipliers in ${\mathbb R}^2$ (3).

121009-092523: Lagrange multipliers in ${\mathbb R}^2$ (2).

121009-092511: Lagrange multipliers in ${\mathbb R}^2$.

121009-091240: The isoperimetric inequality.

121005-094811: The brachistochrone, again.

121005-094413: Conservation of energy (2).

121005-094106: Conservation of energy.

121005-092726: Conservation of momentum.

121005-092109: Reminder of Euler-Lagrange.

121005-090823: Notes and riddles.

121002-103950: Properly writing Euler-Lagrange and the brachistochrone.

121002-103247: $F=ma$ (2).

121002-102822: Deriving Euler-Lagrange (5), $F=ma$.

121002-102506: Deriving Euler-Lagrange (4).

121002-101904: Deriving Euler-Lagrange (3).

121002-101423: Deriving Euler-Lagrange (2).

121002-095520: Deriving Euler-Lagrange.

121002-095243: Example: The brachistochrone.

121002-094521: Example: Classical mechanics.

121002-094030: Example: Power lines.

121002-092726: The basic calculus of variations problem.

121002-092032: Back to the chain rule (2).

121002-091704: Back to the chain rule.

121002-090739: Today's riddle.

121001-095749: Calculus of variations (2).

121001-095740: Higher order equations, calculus of variations.

121001-094905: The fundamental theorem: higher order equations.

121001-094116: The fundamental theorem: systems (2).

121001-093509: The fundamental theorem: systems.

121001-093010: The fundamental theorem: uniqueness (2).

121001-092946: The fundamental theorem: uniqueness.

121001-091541: Review of the fundamental theorem.

121001-090909: Computing $(x^x)'$.

120928-095846: The Fundamental Theorem: Uniform Convergence.

120928-094804: The Fundamental Theorem: $\phi_n-\phi_{n-1}$ is well-bounded (2).

120928-094102: The Fundamental Theorem: $\phi_n-\phi_{n-1}$ is well-bounded.

120928-093027: The Fundamental Theorem: $\phi_n$ is well-defined.

120928-092238: The Fundamental Theorem: the $y'=y$ example.

120928-091714: The Fundamental Theorem: Statement.

120925-095319: The Fundamental Theorem (3).

120925-094906: The Fundamental Theorem (2).

120925-094303: The Fundamental Theorem.

120925-093754: The Lipschitz Condition.

120925-092443: Wishful thinking (3).

120925-092029: Wishful thinking (2).

120925-091230: Wishful thinking.

120925-090721: Riddle Along.

120924-095914: Integrating factors (3).

120924-095800: Integrating factors (2).

120924-095319: Integrating factors.

120924-095019: Exact equations (5).

120924-094422: Exact equations (4).

120924-093936: Exact equations (3).

120924-093412: Exact equations (2).

120924-092906: Exact equations.

120924-092111: Partial derivatives commute (2).

120924-091545: Partial derivatives commute.

120924-090840: Show and tell (2).

120924-090831: Show and tell.

120921-095745: Notes for September 21 (6).

120921-095252: Notes for September 21 (5).

120921-095245: Notes for September 21 (4).

120921-093121: Notes for September 21 (3).

120921-092710: Notes for September 21 (2).

120921-092353: Notes for September 21.

120918-095852: Homogeneous Equations (2).

120918-095416: Homogeneous Equations.

120918-094717: Autonomous Equations (2).

120918-094527: Autonomous Equations.

120918-093915: Changing source and target coordinates (2).

120918-093421: Changing source and target coordinates.

120918-091850: Escape Velocities (2).

120918-091728: Escape Velocities.

120918-090713: Riddle Along.

120917-095813: Escape velocities (2).

120917-095323: Escape velocities.

120917-094437: Separable equations: the easy to justify way (2).

120917-093948: Separable equations: the easy to justify way.

120917-093027: Separable equations: the easy to remember way.

120917-091853: The general problem, separable equations.

120914-095607: First order linear, non-homgeneous (5).

120914-095435: First order linear, non-homgeneous (4).

120914-095224: First order linear, non-homgeneous (3).

120914-094700: First order linear, non-homgeneous (2).

120914-093953: First order linear, non-homgeneous.

120914-093309: First order linear homogeneous (2).

120914-092628: First order linear homogeneous.

120914-091936: $y'=f$ and first order linear homogeneous.

120914-090342: Read along and riddle along.

120911-094639: This is a cycloid (2).

120911-093941: This is a cycloid.

120911-093629: Solving the equation (2).

120911-093043: Solving the equation.

120911-092453: Brachistochrone review.

120910-100221: Deriving the brachistochrone equation (2).

120910-095542: Deriving the brachistochrone equation.

120910-094435: Fermat's principle and Snell's law.

120910-093559: The Brachistochrone problem.

120910-092814: A messy example.

120910-092025: What's a differential equation?