Second-Order Linear Differential Equations
Stub. General and particular solutions of linear equations, linear independence and the Wronskian, nonhomogeneous equations, constant-coefficient homogeneous equations and differential operators, and complex roots of the characteristic equation.
The linear structure theorem for second-order equations: the general solution of a homogeneous linear equation is a combination of two linearly independent solutions (checked via the Wronskian), and the general solution of a nonhomogeneous equation is any particular solution plus the homogeneous general solution. Specializes this to constant-coefficient equations, where the characteristic equation and the differential-operator viewpoint reduce solving to algebra — including the case of complex roots, which produces oscillatory solutions and sets up the vibrations module.
Outline (TODO — flesh out each)
- Intro to second-order linear equations
- General solutions of linear equations
- Linear independence and general solutions — the Wronskian
- Nonhomogeneous linear equations
- Constant-coefficient linear homogeneous equations
- Constant-coefficient differential operators
- Complex roots of the characteristic equation