Second order ODEs with nonconstant coefficients Flashcards Preview

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Flashcards in Second order ODEs with nonconstant coefficients Deck (19)
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1

Describe how to obtain a characteristic polynomial

Replace all derivatives of y with the corresponding power of λ.

 

Note that this is obtained by replacing y(x) with eλx

2

Describe the roots of the characteristic equation

λ1 and λ2 are roots of the characteristic polynomial such that the characteristic equation is obtained. 

3

Give the general solution to the characteristic polynomial if there are two real, distinct roots (ie if the discriminant > 0)

4

Give the general solution to the characteristic polynomial if there is one real, distinct root (ie if the discriminant = 0)

5

Give the general solution to the characteristic polynomial if there are two complex roots (ie if the discriminant

6

Give Euler's formula 

7

Give the expression for a power series centred around x0

8

Give the symbol for natural numbers 

9

Give the symbol for rational numbers 

10

Give the symbol for real numbers 

11

Give the symbol for complex numbers 

12

Give the symbol for integers

13

Describe the Leibniz-Maclaurin method of solving a differential equation with non constant coefficients 

  • Guess a solution as a power series 
  • Work out the appropriate differentials 
  • Plug these into the original DE

14

State three properties of power series 

  1. If a power series can be written as a power series centred around x0 for a some range of x, this power series is unique
  2. Power series can be added/subtracted termwise
  3. Power series can be differentiated termwise

15

Describe the uniqueness property of power series 

If the equation below holds, an must = 0 for all n.

16

Show a general differentiation of a general power series 

17

Define an ordinary point

A point x0 is called an ordinary point if BOTH limits exist for the limits shown, for the equation P(x)y'' + Q(x)y' + R(x) = 0 

18

Define a singular point

A point x0 is described as singular if it cannot be described as ordinary 

19