| Title | Summary |
|---|---|
| Second-Order Linear Equations: Concepts | Paul's On-Line Notes introducing second-order linear equations. They rapidly specialize to the constant coefficient, homogeneous case. They present how to construct general solutions by superposition. They show how to solve initial-value problems from a general solution. They show how to construct special solutions from real roots of the characteristic polynomial. |
| Second-Order Linear Equations: Real, Distinct Roots | Paul's On-Line Notes on second-order homogeneous linear equations with constant
coefficients whose characteristic polynomial has distinct real roots. Five examples are worked. |
| Second-Order Linear Equations: Complex Roots | Paul's On-Line Notes on second-order homogeneous linear equations with constant
coefficients whose characteristic polynomial has a conjugate pair of roots. Four examples are worked. |
| Second-Order Linear Equations: Repeated Roots | Paul's On-Line Notes on second-order homogeneous linear equations with constant
coefficients whose characteristic polynomial has a double real root. The derivation of the recipe uses order reduction, which is not as direct as the one in our text based on the key identity. Three examples are worked. |
| Higher-Order Homogeneous Linear Equations with Constant Coefficients | Paul's On-Line Notes on higher-order homogeneous linear equations with constant
coefficients. The present the general recipe without much justification. Five examples are worked. |
| Complex Numbers and Complex Exponentials | Professor Mattuck (MIT video) gives a review of complex numbers and an introduction to the complex exponential. These enter our course at this point. |
| Solving Second-Order Linear ODEs with Constant Coefficients | Professor Mattuck (MIT video) solves some second-order linear
equations with constant coefficients. He treats the case of two real roots and begins the case of a conjugate pair of roots. |
| Continuation: Complex Characteristic Roots | Professor Mattuck (MIT video) solves some second-order linear equations with constant coefficients. |
| Khan Academy: Second-Order Linear Homogeneous Differential Equations 2 | Khan Academy video showing how to find a general solution of some second-order linear homogeneous equations with constant coefficients when the characteristic polynomial has two real roots. The roots are found using the quadratic formula, while similar examples in the text find them by completing the square. The main point in both this video and the text is that the only difficult step in finding a general solution is finding the roots of the characteristic polynomial. |
| Khan Academy: Second-Order Linear Homogeneous Differential Equations 3 | Khan Academy video showing how to solve an initial-value problem for a second-order linear homogeneous equation with constant coefficients when the characteristic polynomial has two real roots. The problem reduces to solving a linear algebraic system for the values of the two parameters in the general solution. |
| Khan Academy: Second-Order Linear Homogeneous Differential Equations 4 | Khan Academy video showing how to solve another initial-value problem for a second-order linear homogeneous equation with constant coefficients when the characteristic polynomial has two real roots. |
| Khan Academy: Complex Roots of the Characteristic Polynomial 1 | Khan Academy video showing how to find a general solution of a second-order linear homogeneous equation with constant coefficients when the characteristic polynomial has a conjugate pair of roots. The discussion is general at first. The complex roots are found by using the quadratic formula rather than by completing the square. Like our text, it uses the Euler formula to obtain real solutions. (Euler's nane is mispronounced "Ew-ler" rather than "Oy-ler".) |
| Khan Academy: Complex Roots of the Characteristic Polynomial 2 | Khan Academy video showing how to find a general solution of a second-order linear homogeneous equation with constant coefficients when the characteristic polynomial has a conjugate pair of roots. |
| Khan Academy: Complex Roots of the Characteristic Polynomial 3 | Khan Academy video showing how to solve an initial-value problem for a second-order linear homogeneous equation with constant coefficients when the characteristic polynomial has a conjugate pair of roots. The problem reduces to solving a linear algebraic system for the values of the two parameters in the general solution. |
| Khan Academy: Repeated Roots of the Characteristic Polynomial | Khan Academy video showing how to find the general solution of a second-order linear homogeneous equation with constant coefficients when the characteristic polynomial has a double real root. The approach to finding the "second solution" uses the order reduction method, which is far more involved than the approach taken in our text. It also does not extend nicely to higher-order equations. |
| Khan Academy: Repeated Roots of the Characteristic Polynomial Part 2 | Khan Academy video showing how to solve an initial-value problem for a second-order linear homogeneous equation with constant coefficients when the characteristic polynomial has a double real root. The problem reduces to solving a linear algebraic system for the values of the two parameters in the general solution. |