Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Maxwell's equations wikipedia , lookup
BKL singularity wikipedia , lookup
Equations of motion wikipedia , lookup
Differential equation wikipedia , lookup
Navier–Stokes equations wikipedia , lookup
Computational electromagnetics wikipedia , lookup
Exact solutions in general relativity wikipedia , lookup
Homog. Equations with Constant Coefficients Consider ay + by + cy = 0. A solution will have the form y = emx This is called the auxiliary equation. 3 Cases If distinct roots m1 and m2 y c1e m1 x If repeating root m y c1e c2 xe mx If complex conjugate roots α ± βi x ye c1 cos x c2 sin x c2e mx m2 x Ex. Solve 2y – 5y – 3y = 0 Ex. Solve y – 10y + 25y = 0 Ex. Solve y + 4y + 7y = 0 Ex. Solve the IVP 4y + 4y + 17y = 0, y(0) = –1, y (0) = 2 Useful to remember: y + k2y = 0 has the solution y c1 cos kx c2 sin kx This works for higher order equations, but it’s harder to break down the cases: If all roots are distinct m1, m2, …, mn mn x m1 x m2 x y c1e c2e cne If a root m repeats k times mx mx 2 mx y c1e c2 xe c3 x e k 1 mx ck x e Remember, if there is a complex root, then its conjugate is also a root: α ± βi A cubic polynomial must have at least one real root. Ex. Solve y + 3y – 4y = 0 Ex. Solve y + 2y + y = 0 Ex. Solve y – 3y = 0