# Part 2: Theory for second-order linear equations, general solutions for homogeneous equations with constant coefficients, the method of undetermined coefficients, electrical and mechanical oscillatory systems, and the Laplace transform xml version="1.0" encoding="UTF-8" standalone="no"? --- 1. What can be said generally about higher order **linear** equations? * [11.  Higher Order Linear Equations - Theory *](https://www.menti.com/alfbop5egtvr) 3. How do we solve problems when the homogeneous problem is also autonomous? * [12. Homogeneous solutions](https://www.menti.com/ali7edesjk9k) 5. How does the method of undetermined coefficients work for constant coefficient problems with inhomogeneous terms? * [13. The Method of Undetermined Coefficients (2nd Order)](https://www.menti.com/algkttgiejhq) 7. What are the contexts for this mathematical symbolism? What are beats and resonance? * [14. Modeling Simple Oscillatory Dynamics](https://www.menti.com/al18vk8vz2q8) 9. What is the Laplace transform? * [15. Laplace Transform Definition](https://www.menti.com/al6qymmqguak) * [16. Algebra Review](https://www.menti.com/al7d4idxo1nc) * [17. Inverse Transforms and Transforms of Derivatives](https://www.menti.com/alsyf4uexkh3) 11. How do we use the Laplace transform for time-invariant linear systems, both driven and un-driven? * [18. Operational Properties and UnitStep](https://www.menti.com/algysk3xm655) 13. What is the unit step-function and how does it Laplace transform? * [18. Operational Properties and UnitStep](https://www.menti.com/algysk3xm655) 15. What is the Dirac delta function and how does it Laplace transform? * [19. The Dirac delta function and LT](https://www.menti.com/al8g6c2ti5qi) --- ### Part 2.1: Definitions, theory, constant coefficient solution forms, and the method of undetermined coefficients 1. Higher-order linear equations, linear combinations, linear independence and the Wronskian determinant * [Discussion of linear combination and linear independence - Part 1 (9min)](https://youtu.be/IyVqs1UuGAk) * [Discussion of linear combination and linear independence - Part 2 (10min)](https://youtu.be/T383HRvd2bU) * [IVP v. BVP (13 min)](https://youtu.be/nhmOHb3ZWT8) 2. Second-order linear autonomous homogeneous problems, exponential functions, algebraic root finding, and the general solution * [2nd Order Constant Linear Homogeneous Equation: Real Different Roots - Example 1: y''+5y'+ 6y=0 (8min)](https://youtu.be/GZDp8-TsbSk) * [2nd Order Constant Linear Homogeneous Equation: Real Different Roots - Example 2: y''+y'- 2y=0 (4min)](https://youtu.be/SwrYVpUBZcY) * [2nd Order Constant Linear Homogeneous Equation: Repeated Roots - Example 2: y''+4y'+4y=0 (13min)](https://youtu.be/DidHaJ0wenE) * [2nd Order Constant Linear Homogeneous Equation: Complex Roots - Example 1: y''+4y'+13y=0 (17min)](https://youtu.be/XOYEmkrsou8) 3. The method of undetermined coefficients for second-order linear ODE with constant coefficients. * [MUC Redux Intro](https://youtu.be/2Ze6aRstFT0) (5mins) * [Review of homogeneous forms and MUC table](https://youtu.be/pNsVRYFm_YQ) (3mins) * [Real Roots with Constant Inhomogeneous Term](https://youtu.be/H8rw91Rj4-I) (8mins) * [MUC multiple choice questions and additional MUC Problems](https://youtu.be/M8mlWz40I5Q) (20mins) * [Homogeneous Repeated Root Case, MUC and x multiplication](https://youtu.be/PxWrrr1xubs) (21mins) * [MUC and Resonance](https://youtu.be/_lzL2ZwxcAU) (9mins) * Fall2020 Repo + [The Method of Undetermined Coefficients](https://www.youtube.com/watch?v=XQoFKOcr_y4) (5mins - Robot Scott) + [The Method of Undetermined Coefficients - Special Case](https://www.youtube.com/watch?v=tqLnVGLSmCM) (8mins - Robot Scott) - This discusses the t-multiplication problem when the particular solution is proportional to the homogeneous solution + [MATH235 - Method of Undetermined Coefficients](https://www.youtube.com/watch?v=X097hiB1wyY&list=PLfzyv9XksyEdYDYQ2XNt-EHEcdnKD4YWN) (youtube playlist - problem examples - first order) + [Solving y'+2y=3, y(0)=1, via the Method of Undetermined Coefficients](https://www.youtube.com/watch?v=-fxkT8xckBY) * + [Solving y'-3y = 2t^2 via the Method of Undetermined Coefficients](https://www.youtube.com/watch?v=w4awjM3SVn0) * + [Solving y' +2y=3+4e^5t via the Method of Undetermined Coefficients](https://www.youtube.com/watch?v=iSpoHnhM4-s) * + [Solving y'+5y = 3cos(2t) via the Method of Undetermined Coefficients](https://www.youtube.com/watch?v=Lpx77ZOtt1g) * + [Solving y'+2y=3e^t via the Method of Undetermined Coefficients](https://www.youtube.com/watch?v=C6bYgTkbpjA) * --- ### Part 2.2: Mathematical modeling with second-order linear equations and mass-spring dynamics 1. **Mass-Spring Dynamics**: An object with constant mass, attached to an ideally elastic spring, subjected to linear drag, and an inhomogeneous external force, i.e., the classical mass-spring system. * [Discussion of the forces leading to dynamics as it relates to the initial value problem](https://youtu.be/C3BNzpNl8MA). (2 mins) + [Derivation of Mass-Spring System and Discussion of Solution Structure](https://youtu.be/oQ2F0rK7Q90) (18m) [F20 Resource] * [Derivation of an initial value problem from a word problem given in imperial units.](https://youtu.be/zrpXqxFfV44) (12 mins) * [Discussion of simple harmonic motion and damping types w/ commentary on a pendulum experiment](https://youtu.be/30a-gNwjh8k) (14 mins) + [Homogeneous Mass-Spring Systems and the Role of Damping](https://youtu.be/gFBssZE506I) (10m) [F20 Resource] + Reference video: [Critical Damping -- xmdemo 068](https://www.youtube.com/watch?v=99ZE2RGwqSM) * [Discussion of the role of drag, visualizations of the system, and commentary on coin-funnel and tube television](https://youtu.be/W0DbXIRXm3w) (18 mins) * [Derivation of resonance and commentary on resonance in complicated structures exhibiting simple oscillations](https://youtu.be/A0yyD_lVd-Q) (24 mins) + [Forced Mass Spring Systems - Resonance and Beats](https://youtu.be/FKYB3EfjT60) (38m) [F20 Resource] + Reference video: [MIT Physics Demo Tuning Forks Resonance and Beat Frequency](https://youtu.be/pRpN9uLiouI) + Reference video: [London Millennium Bridge Opening](https://www.youtube.com/watch?v=gQK21572oSU) + Reference video: [Millennium Bridge(Raw Footage)](https://youtu.be/eAXVa__XWZ8) + Reference video: [Breaking Glass with Sound](https://www.youtube.com/watch?v=CdUoFIZSuX0) + Reference video: [What is an inductor?](https://www.youtube.com/watch?v=NgwXkUt3XxQ) + Reference video: [Resonance and Tesla Coil Basics](https://www.youtube.com/watch?v=fT5_MmeSeDU) + Reference video: [Top 7 Game Songs on TESLA COILS](https://www.youtube.com/watch?v=9PLa5gW_FF4) --- ### Part 2.3: The Laplace transform, solving IVP with the Laplace transform, the unit-step function, and the Dirac delta function 1. Laplace Transform Definition, Working with the Laplace Transform Table, Inverse Transforms and Transforms of Derivatives * [The Definition of the Laplace Transform](https://youtu.be/UHpnMEi669U) (4mins) * [Laplace transform of a piecewise function through the definition](https://youtu.be/jmB8oUkUZss) (8mins) * [The Laplace transform of an exponential function by the definition](https://youtu.be/lda6aCWmiqo) (5mins) * [The Laplace transform of t e^(-t) by the definition](https://youtu.be/3S4OldG9lhM) (6mins) * [The Laplace transform through a table](https://youtu.be/ah4z2axkoZI) (5 mins) * [Introduction to Laplace transforms and the Laplace transform table](https://youtu.be/CSjdVMoBbck) (6mins) * [Laplace transform of the derivative by the definition](https://youtu.be/dQ88d8MH1pE) (8mins) * [Inverse Laplace transforms and the table with some algebraic tricks](https://youtu.be/zwZh7srg65A) (9mins) * [Inverse Laplace transform problem with a significant partial fraction step](https://youtu.be/jLv3ILKdpzQ) (16mins) * [Inverse Laplace transform problem tied to y'' + 5 y' + 6 y= 0, y(0)=0, y'(0)=1](https://youtu.be/GeDk3HWxipE) (7mins) 2. Laplace transforms and Initial Value Problems * [Laplace transform solution to y' = a y, y(0)=1](https://youtu.be/u3FN3lBOzto) (8mins) * [Summary of Laplace transforms applied to second order linear ODE with constant coefficients](https://youtu.be/fUlCRoRPlns) (8mins) * [Laplace transform solution y''-4y' = 3e^2t -6e^-t, y(0)=1, y'(0)=-1](https://youtu.be/WZBZO4SiSo0) (15mins) * [A bit about the Laplace variable s, as it relates to initial value problems](https://youtu.be/6_8K9CzW1iY) (2mins) * [Laplace transform solution to y'' + 5 y' + 6 y= 0, y(0)=0, y'(0)=1](https://youtu.be/_CBNFXHIpAs) (10mins) * [Laplace transform solution to complex root problem y'' - y' + y = 0, y(0)=2, y'(0)=5](https://youtu.be/UcKMBQLK-Kc) (18mins) * [Laplace transform solution outline to y' + y = cos(2t), y(0)=2 highlighting partial fraction decomp](https://youtu.be/vbEIdVES7-c) (5mins) * [Laplace transforms and second order linear constant coefficient IVP based on root types](https://youtu.be/KXa7fTSYlcA) (12mins) * [Laplace transforms and repeated roots y'' + 4 y' + 4 y = 6e^(-2t), y(0) = -1, y'(0)=5](https://youtu.be/N3HdEwJcjc0) (20mins) + [Partial fraction work for y'' + 4 y' + 4 y = 6e^(-2t), y(0) = -1, y'(0)=5](https://youtu.be/cWQX2_CjWA4) (5mins) * Fall 2020 resources associated with the three homogeneous forms + [Laplace Transform of second order constant linear homogeneous IVP with real different roots](https://youtu.be/FkJP7FG9QSs) (17min) [F20 Resource] + [Thinking Through Solving an IVP w/ LT (Real different roots and partial fraction decomposition)](https://youtu.be/UMu7KV4nXcg) (11min) [F20 Resource] + [Solving y'' - y' + y = 0, y(0)= 2, y'(0)=-5 with Laplace transforms](https://youtu.be/B4U5_BWKUY0) (20min) [F20 Resource] 3. Unit-step function, Dirac delta "function," definitions, transforms and initial value problems * [Definition of the unit-step function](https://youtu.be/0k5wrjX0CIk) (5mins) * [Representing a given piecewise function through step functions](https://youtu.be/HRWl7PB2VQY) (9mins) * [Step Function (Part 1) - Definition and Usage](https://www.youtube.com/watch?v=DmeuNQ4Qr7Q) (12min) * [Step Functions (Part 2) - Review, Forward, and Inverse Transformations](https://youtu.be/eyYLP4FXOM0) (32min) * [Step Functions (Part 3) - Justification of Step Function Table Entries and Usage](https://youtu.be/Ml4FGCqG_0o) (30min) [F20 Resource] * [Step Functions (Part 4) - Solving y'+y = u(t-1), y(0)=0 with Laplace transforms](https://youtu.be/XMktu333_dk) (11min) * [Explanation of the Dirac delta function and solving an IVP with unit steps and delta functions](https://youtu.be/vS8Hsmh7rn8) (50min) * [Review of the unit-step function and its Laplace transform](https://youtu.be/tJYWkIlXOBY) (5mins) * [The properties of the Dirac delta "function"](https://youtu.be/esPCphQ_E-4) (10mins) * [Laplace transform solution to y'' + 5y' + 6y = deltat 2, y0=y'0=0](https://youtu.be/uN3d3r44tCA) (13mins) * [Laplace transform solution to y'' + 4y' + 13y = deltat pi, y0=0, y'0=1](https://youtu.be/joumPM1hnZc) (11mins) * Archive of Old Resources + [Laplace Transform: Integral definition, properties and a simple IVP](https://youtu.be/JzDGUKv_WeU) (8mins) + [Laplace Transform: Second Order Linear Initial Value Problem w/ Constant Coefficients Three ways](https://youtu.be/-4vI3fja-jw) (17mins) + [Laplace Transforms: Relationship between Dirac and Unit-Step via Energy of Mass-Spring Equation](https://youtu.be/qDlrnOQlZOs) (5mins) + [Laplace Transform: The Unit Step Function](https://youtu.be/yia7bUqP7GA) (4mins) + [Laplace Transforms: Dirac "function"](https://youtu.be/eHiOmSPJE3A) (5mins) + [Laplace Transform: Transforms of Dirac and Unit Step Function w/ Application to Mass-Spring](https://youtu.be/NA_beFdDTDQ) (13mins) 4. Algebra Resources + Reference video: [Partial Fraction Decomposition - Example 1 [Linear factors]](https://www.youtube.com/watch?v=HZTv4zCgEnA) (6min) + Reference video: [Partial Fraction Decomposition - Example 5 [Repeated Linear Factors]](https://youtu.be/Slfrb8aEj08)(11min) + Reference video: [Long Partial Fractions Problem - Repeated Irreducible Quadratic Factors, Part 1](https://youtu.be/WqKgvMzKICM)(13min) + Reference video: [Completing the Square - Solving Quadratic Equations](https://youtu.be/xGOQYTo9AKY)(4min)