differential equations

Question 1

independent variable

Answer 1

derivates appear with respect to this quantity

Question 2

dependent variable

Answer 2

specificed by the independent variable - derivatives of the dependent variable with respect to the independent variable are rates of change
a solution of the ODE specifies the fucntion rul for the dependent variable in terms of the independent variable

Question 3

order

Answer 3

the highest derivative of the dependent variable

Question 4

autonomous

Answer 4

do not explicitly involve the independent variable and are of the form
R(y, dy/dx, dy^2/dx^2, …) = 0

Question 5

linear

Answer 5

equations have the dependent variable and its derivates appearing to the first degree only
i.e. no products, powers, or transcendental functions of the dependent variable

Question 6

homogeneous

Answer 6

linear differential equations have the zero function y(x) = 0 for all x as a solution

Question 7

constant coefficient

Answer 7

linear differential equations have fixed coefficients of the all the dependent variable terms

Question 8

what is the difference between initial and boundary conditions

Answer 8

initial conditions are specified for the same value of the independent variable
boundary conditions are specified for different values of the independent variable

Question 9

homogenous functions

Answer 9

if all its arguments are multiplied by a factor, then its value is multiplied by some power of this facter
f(ax) = a^k f(x)
for some constant a and integer k