Partial differential equations

Question 1

Define an ordinary differential equation

Answer 1

An ordinary differential equation is one in which the unknown function is a function of only one variable eg:

Question 2

Define a partial differential equation

Answer 2

A partial differential equation is one in which the function is a function of more than one variable

Note: it is important to note whether or not these variables are independent of one another. If not, it is necessary to write which one is to be held constant

Question 3

Define the order of a differential equation

Answer 3

The order of a DE is the order of the highest derivative

Question 4

Define a linear differential equation

Answer 4

A D.E is linear if the unknown function does not appear in any power higher than 1

Question 5

Define a homogeneous differential equation

Answer 5

A linear differential equation is homogeneous if all the terms depend on the unknown function or its derivatives. For the example below, the DE is homogeneous if G(x) = 0

Question 6

Define an inhomogeneous differential equation

Answer 6

A linear differential equation is inhomogeneous if not all the terms depend on the unknown function or its derivatives. For the example below, the DE is homogeneous if G(x) =/= 0

Question 7

Define the degree of a differential equation

Question 8

Describe the method of separation of variables

Answer 8

• The aim of separating a differential equation with n variables is to form n separate differential equations.
• For a DE with 2 variables x,y we can set u(x,y) = X(x)Y(y) and substitute this into the original equation.
• We can then change the partial derivatives into ordinary differentials as X and Y only depend on one variable each.
• Divide through by XY and rearrange to get X on one side and Y on the other
• x and y are independent variables so the separate DEs must both equal the same consant