Calculus Volume 3 · free preview
Second-Order Linear Equations
Second-order linear equations with constant coefficients solve through the characteristic equation, the roots' nature giving exponentials, sines, and cosines, the general solution a two-parameter family matching the two initial conditions.
1. The homogeneous equation and the characteristic equation
The homogeneous equation and the characteristic equation
2. The three root cases
The three root cases
3. Initial value problems
Initial value problems
4. The Wronskian and the fundamental solutions
The Wronskian and the fundamental solutions
5. Characteristic equation
The equation y double prime minus y = 0 is solved by which characteristic equation?
6. Complex roots solution
Complex characteristic roots alpha plus or minus i beta give which solution form?
7. Repeated root solution
A repeated characteristic root r gives which general solution?
8. Two initial conditions
A second-order equation's general solution has two arbitrary constants because of which fact?
9. Exponential ansatz rationale
Why does the exponential e^(rt) solve constant-coefficient linear equations?
10. Harmonic oscillator frequency
The oscillator y double prime plus omega squared y = 0 has angular frequency omega. As a single letter, what is the frequency parameter? Answer with one lowercase letter.
11. Wronskian definition
The Wronskian of y1 and y2 is y1 y2 prime minus which product? Answer like y1 prime y2.
12. Discriminant classification
The characteristic's discriminant classifies the roots' cases. As two letters, what is it? Answer like b^2 - 4ac.
13. Superposition principle
If y1 and y2 solve the homogeneous equation, which combination also solves it?
14. Spring mass natural frequency
The spring-mass system's natural frequency is the root of k over m. As a fraction, what is under the root? Answer like k/m.
Practice this interactively
Free account · instant grading · spaced review that schedules itself.
Start this course for free