Ordinary Differential Equations

Question 1

What is the integrating function of a first order differential equation?

Answer 1

IF = e^(integral(b(x)) or IF = exp(int(b(x)))

Question 2

What is the difference between a general solution and a particular solution of an ODE?

Answer 2

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.

Question 3

What are boundary conditions on an ODE?

Answer 3

Additional conditions that lead to a particular solution.

Question 4

In general, how can ODEs be written?

Answer 4

F(x, y, dy/dx, d^2y/dx^2, …) = 0.

Question 5

What value defines the order of an ODE?

Answer 5

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.

Question 6

What is a directly integrable ODE?

Answer 6

An ODE which fits the form dy/dx = f(x).

Question 7

How do you solve a seperable first-order ODE (in form dy/dx = g(x)h(y))?

Answer 7

• 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)

Question 8

How must the right hand side of a first-order ODE be written in order to have a homogenous ODE?

Answer 8

If the variables x and y appear in the ratio y/x then the ODE is said to be homogenous.

Question 9

What is the first step in solving a homogenous ODE?

Answer 9

You create a new dependant variable

v(x) = y(x)/x, or y(x) = x.v(x)

Question 10

What type of ODEs use the Integrating Factor in order to solve?

Answer 10

First-order linear ODEs.

Question 11

A first order ODE (with independant variable t) of the form dy/dt = f(y) is said to be ____.

Answer 11

Autonomous.

Question 12

What are the zeros of f(y) in the ODE

dy/dt = f(y)

are called the _____ of the ODE?

Answer 12

The equilibrium or critical points of the ODE.

Question 13

What is the complementary function of a second order, linear, non-homogenous equation?

Answer 13

The general solution of the homogenous equation.

Question 14

How do the complementary function and the particular integral of a 2nd order, linear non-homogenous equation?

Answer 14

Where Ygs = General Solution, Ycf = Complementary Function and Ypi = Particular integral,

Ygs = Ycf + Ypi

Question 15

What is the Particular Integral of a 2nd order linear non-homogenous equation?

Answer 15

Any solution of the ODE, normally determined by the nature of f(x).

Question 16

In what case is a Particular Integral of ODE F containing term g(x) unsuitable to use, and how may you solve this?

Answer 16

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.

Question 17

What is an auxillary equation and on what kind of ODE is it used?

Answer 17

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.

Question 18

How do you handle complex conjugate roots of an auxillary equation?

Answer 18

When the roots satisfy M1,2 = α + iβ, the Complimentary Function will be of form

Ycf = e^αx[Acos(βx) + Bsin(βx)]

Question 19

What is the form of Complimentary function when the roots of the auxillary equation, M1,2, are equal? (i.e M1 = M2)

Answer 19

The Complimentary Function will be in the form

Ycf = [A + Bx]e^mx.