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 e^{λx}

Describe the roots of the characteristic equation

λ_{1 }and λ_{2} 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

Give Euler's formula

Give the expression for a power series centred around x_{0}

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

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

State three properties of power series

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- If a power series can be written as a power series centred around x
_{0} for a some range of x, this power series is unique
- Power series can be added/subtracted termwise
- Power series can be differentiated termwise

_{0}for a some range of x, this power series is uniqueDescribe 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 x_{0 }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 x_{0} is described as singular if it cannot be described as ordinary