Ordinary differential equations Flashcards Preview

PHAS0009 > Ordinary differential equations > Flashcards

Flashcards in Ordinary differential equations Deck (15)
Loading flashcards...
1

Define a linear ODE

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

2

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 

3

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. 

4

Describe how to find the general solution of a separable ODE

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

5

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. 

6

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

 

7

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

8

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

9

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

10

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

Substituting y = ekx gives k2 + pk + q = 0

Specific solutions are y1 = ek1x and y2 = ek2x

General solutions depend on the nature of k1 and k2

11

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

12

Give the general solution to a second-order homogeneous linear ODE with constant coefficients if it has complex roots, k1,2 = α ± iβ

13

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

14

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

15

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

First set f(x) = 0 and find the 'complementary function yCF(x)' of the homogeneous ODE. 

Find a particular integral yPI(x), then y(x) = yCF(x) + yPI(x)