Second order ODEs with nonconstant coefficients Flashcards
(19 cards)
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
Describe the roots of the characteristic equation
λ1 and λ2 are roots of the characteristic polynomial such that the characteristic equation is obtained.
Give the general solution to the characteristic polynomial if there are two real, distinct roots (ie if the discriminant > 0)
Give the general solution to the characteristic polynomial if there is one real, distinct root (ie if the discriminant = 0)
Give the general solution to the characteristic polynomial if there are two complex roots (ie if the discriminant < 0)
Give Euler’s formula
Give the expression for a power series centred around x0
Give the symbol for natural numbers
Give the symbol for rational numbers
Give the symbol for real numbers
Give the symbol for complex numbers
Give the symbol for integers
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
State three properties of power series
- 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
- Power series can be added/subtracted termwise
- Power series can be differentiated termwise
Describe the uniqueness property of power series
If the equation below holds, an must = 0 for all n.
Show a general differentiation of a general power series
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
Define a singular point
A point x0 is described as singular if it cannot be described as ordinary
