Rather than giving a strict mathematical definition, let us look at an example of a PDE, the heat equation in 1 space dimension
(1.5) |
(1.7) |
is called inhomogeneous, due to the term on the right, that is independent of .
Why is all that so important? A linear homogeneous equation allows superposition of solutions. If and are both solutions to the heat equation,
(1.8) |
any combination is also a solution,
(1.9) |
For a linear inhomogeneous equation this gets somewhat modified. Let be any solution to the heat equation with a inhomogeneity,
(1.10) |
In that case , with a solution to the homogeneous equation, see Eq. ( 1.8 ), is also a solution,
Finally we would like to define the order of a PDE as the power in the highest derivative, even it is a mixed derivative (w.r.t. more than one variable).
Quiz Which of these equations is linear? and which is homogeneous?
What is the order of the following equations?