Homogeneous Linear Ordinary Differential Equations

In this video we discuss how to solve homogeneous linear ordinary differential equations (ODEs). The approach outlined in this lecture is applicable to higher order ODEs but we focus on an example of

a*x’’(t) + b*x’(t) + c*x(t) = 0

with x(0) = x0 and x’(0) = xdot0

We discuss how to find a general solution to the aforementioned ODE and then how to choose coefficients of the solution to satisfy the initial conditions. This process is sometimes referred to as the traditional approach to solving homogeneous linear ODEs. We compare how the Laplace technique yields the same solution.

Topics and timestamps:
0:00 – Introduction
01:16 – Linear ODE definition
06:10 – General solution
14:20 – Case 1: real distinct roots
38:21 – Case 1 using the Laplace technique
46:23 – Case 2: repeated real roots
55:43 – Case 3: complex conjugate roots

References:
-Introduction to Ordinary Differential Equations (coming soon…)
-Complex Numbers, Complex Variables, and Complex Functions (links directly to the discussion on Euler’s Formula) (https://youtu.be/WEYX-wa9csU?t=690)
-The Laplace Transform (https://youtu.be/q0nX8uIFZ_k)
-Finding Roots of a Polynomial Using Matlab, Mathematica, and a TI-83 (https://youtu.be/J8il5eB_VS8)
-Partial Fraction Expansion/Decomposition (https://youtu.be/vlCdCAEtRag)
-The Inverse Laplace Transform (https://youtu.be/wZkrU1lPObM)
-Nonhomogeneous Linear Ordinary Differential Equations (coming soon…)

All Ordinary Differential Equation videos in a single playlist (https://www.youtube.com/playlist?list=PLxdnSsBqCrrHHvoFPxWq4l9D93jkCNIFN)

#ODEs

Видео Homogeneous Linear Ordinary Differential Equations канала Christopher Lum