I have to teach engineering mathematics soon. Much of it revolves around the development, manipulation, and solution of differential equations. In this post, I have gathered good resources just for this purpose.
Following are a set of lectures by much-revered professor Gilbert Strang on differential equations.
Overview of Differential Equations
Description: Differential equations connect the slope of a graph to its height. Slope = height, slope = -height, slope = 2t times height: all linear. Slope = (height)2 is nonlinear. Related section in textbook: 1.1 Instructor: Prof. Gilbert Strang
The Calculus You Need
Description: The sum rule, product rule, and chain rule produce new derivatives from known derivatives. The Fundamental Theorem of Calculus says that the integral inverts the derivative. Related section in textbook: 1.2 Instructor: Prof. Gilbert Strang
By the way, I was wondering that why do we multiply with the exponent while taking derivatives (if you remember the power rule). I looked up on the Internet and found this cool explanation on Coursera.
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Response to Exponential Input
Description: For a linear equation with exponential input from outside and exponential growth from inside, the solution is a combination of two exponentials. Related section in textbook: 1.4b Instructor: Prof. Gilbert Strang
Solution for Any Input
Description: To solve a linear first order equation, multiply each input by its growth factor and integrate those outputs. Related section in textbook: 1.4d Instructor: Prof. Gilbert Strang
Step Function and Delta Function
Description: A unit step function jumps from 0 to 1. Its slope is a delta function: Zero everywhere except infinite at the jump. Related section in textbook: 1.4e Instructor: Prof. Gilbert Strang
Response to Complex Exponential
Description: For linear equations, the solution for a cosine input is the real part of the solution for a complex exponential input. That complex solution has magnitude G (the gain). Related section in textbook: 1.5 Instructor: Prof. Gilbert Strang
Integrating Factor for Constant Rate
Home ” Supplemental Resources ” Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler ” Differential Equations and Linear Algebra ” First Order Equations ” Integrating Factor for Constant Rate
Integrating Factor for a Varying Rate
Description: The integral of a varying interest rate provides the exponent in the growing solution (the bank balance). Related section in textbook: 1.6b Instructor: Prof. Gilbert Strang
The Logistic Equation
Description: When competition slows down growth and makes the equation nonlinear, the solution approaches a steady state. Related section in textbook: 1.7 Instructor: Prof. Gilbert Strang
The Stability and Instability of Steady States
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Separable Equations
Home ” Supplemental Resources ” Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler ” Differential Equations and Linear Algebra ” First Order Equations ” Separable Equations
Second Order Equations
Description: For the oscillation equation with no damping and no forcing, all solutions share the same natural frequency. Related section in textbook: 2.1 Instructor: Prof. Gilbert Strang
Forced Harmonic Motion
Description: When the forcing is a sinusoidal input, like a cosine, one particular solution has the same form. But if the forcing frequency equals the natural frequency there is resonance. Related section in textbook: 2.1b Instructor: Prof. Gilbert Strang
Unforced Damped Motion
Description: With constant coefficients in a differential equation, the basic solutions are exponentials. The exponent solves a simple equation. Related section in textbook: 2.3 Instructor: Prof. Gilbert Strang
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