Ordinary Differential Equations Flashcards
What is the integrating function of a first order differential equation?
IF = e^(integral(b(x)) or IF = exp(int(b(x)))
What is the difference between a general solution and a particular solution of an ODE?
A G.S involves arbitrary constants. A P.S takes into account particular circumstances to the equations, and gives specific values to the arbitrary constants.
What are boundary conditions on an ODE?
Additional conditions that lead to a particular solution.
In general, how can ODEs be written?
F(x, y, dy/dx, d^2y/dx^2, …) = 0.
What value defines the order of an ODE?
The highest level of differentiation involved in the equation.
for example, for an equation involving d^4y/dx^4 as its highest level of differentiation, it would be a fourth order ODE.
What is a directly integrable ODE?
An ODE which fits the form dy/dx = f(x).
How do you solve a seperable first-order ODE (in form dy/dx = g(x)h(y))?
- divide both sides by h(y) (1/h(y)dy/dx = g(x))
- integrate both sides w.r.t x to find the general solution of
int(1/h(y) dy) = int(g(x) dx)
How must the right hand side of a first-order ODE be written in order to have a homogenous ODE?
If the variables x and y appear in the ratio y/x then the ODE is said to be homogenous.
What is the first step in solving a homogenous ODE?
You create a new dependant variable
v(x) = y(x)/x, or y(x) = x.v(x)
What type of ODEs use the Integrating Factor in order to solve?
First-order linear ODEs.
A first order ODE (with independant variable t) of the form dy/dt = f(y) is said to be ____.
Autonomous.
What are the zeros of f(y) in the ODE
dy/dt = f(y)
are called the _____ of the ODE?
The equilibrium or critical points of the ODE.
What is the complementary function of a second order, linear, non-homogenous equation?
The general solution of the homogenous equation.
How do the complementary function and the particular integral of a 2nd order, linear non-homogenous equation?
Where Ygs = General Solution, Ycf = Complementary Function and Ypi = Particular integral,
Ygs = Ycf + Ypi
What is the Particular Integral of a 2nd order linear non-homogenous equation?
Any solution of the ODE, normally determined by the nature of f(x).
In what case is a Particular Integral of ODE F containing term g(x) unsuitable to use, and how may you solve this?
If the term g(x) appears in the Complimentary Function of F it is unsuitable.
To resolve this, multiply G(x) by x until this value no longer appears in the CF.
What is an auxillary equation and on what kind of ODE is it used?
An auxillary equation is used in the solution of a 2nd order, linear homogenous equation.
It is found by referring to the ODE as a type of polynomial, by replacing the instances of y with a variable m, to the power of its order of differentiation.
For example,
12(d^2y/dx^2) + 4(dy/dx) - 3y = 0
Would have an auxillary equation of
14m^2 + 4m - 3 = 0.
How do you handle complex conjugate roots of an auxillary equation?
When the roots satisfy M1,2 = α + iβ, the Complimentary Function will be of form
Ycf = e^αx[Acos(βx) + Bsin(βx)]
What is the form of Complimentary function when the roots of the auxillary equation, M1,2, are equal? (i.e M1 = M2)
The Complimentary Function will be in the form
Ycf = [A + Bx]e^mx.
