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Unformatted text preview: Notes for Day : . : Higher Order Homogeneous Constant Coe cient Di erential Equations Another tip: Multiplication and roots of complex numbers
Suppose a + b i occupies the same point on the complex plane as (r , ) occupies on the polar plane, and similarly a + b i occupies the same point as (r , ). en (a + b i)(a + b i) occupies the same point as (r r , + ). (Notice that we multiply the radii, but add the angles. Example:
"Visually" multiply ( + i) (- i). To nd the nth roots of a complex number, determine ( n r, n). Notice that since there are multiple possibilities for , there are multiple possibilities for n. Example :
Find the fourth roots of - . We discussed the theory behind higher-order homogeneous equations at the end of class last time. Today, we'll work through a few of the exercises in the book: p. , y (t) and y (t) are solutions to the initial value problem y + t y + t y = , y ( ) = , y ( ) = - , y ( ) = - , y ( ) = Decide whether y and y form a fundamental set. Example
e solutions , - t - t + , and at + t + cannot form a fundamental set for which value of a? p. ,
y + y + y = , y ( ) = , y ( ) = , y ( ) = - , y ( ) = Suppose we had the initial value problem: e usual fundamental set does not satisfy these initial conditions. Find a fundamental set which does. p. ,
y + y - y - y = . Find the general solution of the di erential equation p. ,
y( ) + y + y = . Find the general solution of the di erential equation ...
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