Define a linear ODE

An ODE is linear if the dependent variable occurs at most to the first power.

Define a homogeneous ODE

A linear ODE is homogeneous if the dependent variable appears to the first power in every term.

For the general form shown below, b(x) must = 0

Describe how to determine whether or not an ODE is separable

Split dy/dx into dy and dx. If the ODE can be separated such that dy and all quantities containing y are on the left and dx and all quantities containing x are on the right, then the ODE is separable.

Describe how to find the general solution of a separable ODE

Integrate the two sides of the equation by their respective variable.

Describe the integrating factor method of solving a linear first-order ODE

S(x) denotes an integrating factor: S(x) = e^{∫P(x)dx }where P is described below in the equation for the standard form of a linear first order ODE.

Multiply all terms by S(x) and integrate both sides to determine an expression for the dependent variable.

Describe the Perfect differential method (also known as the exact differential method) for solving a first order ODE

For an ODE that can be written as P(x,y)dx + Q(x,y)dy = 0, P and Q can be written as they are below.

Find a function f such that f(x,y) = C

Give the necessary condition for P and Q to satisfy in order for the exact differential method to be relevant

Give the standard form of a second-order linear ODE with constant coefficients

Give the standard form of a homogeneous second-order linear ODE with constant coefficients

Describe how to get specific solutions to a second-order homogeneous linear ODE with constant coefficients

Substituting y = e^{kx }gives k^{2} + pk + q = 0

Specific solutions are y_{1} = e^{k1x} and y_{2} = e^{k2x}

General solutions depend on the nature of k_{1} and k_{2}

Give the general solution to a second-order homogeneous linear ODE with constant coefficients if it has two real roots

Give the general solution to a second-order homogeneous linear ODE with constant coefficients if it has complex roots, k_{1,2} = α ± iβ

Give the general solution to a second-order homogeneous linear ODE with constant coefficients if it has degenerate roots

Give the standard form of an inhomogeneous second-order linear ODE with constant coefficients

Describe how to solve an inhomogeneous second-order linear ODE with constant coefficients

First set f(x) = 0 and find the 'complementary function y_{CF}(x)' of the homogeneous ODE.

Find a particular integral y_{PI}(x), then y(x) = y_{CF}(x) + y_{PI}(x)