39. Homogeneous and inhomogeneous linear equations. Liouville formula.

Homogeneous and inhomogeneous linear equations

A differential equation is an equation involving an unknown function of one or more variables, its derivatives and the independent variables. This type of equations comes up often in many different branches of mathematics. They are also especially important in many problems in physics and engineering.

There are many types of differential equations. An ordinary differential equation (ODE) is a differential equation where the unknown function depends on a single variable. A general ODE has the form

where the unknown  f  is usually understood to be a real or complex valued function of x , and x is usually understood to be either a real or complex variable. The order of a differential equation is the order of the highest derivative appearing in Eq. (1). In this case, assuming that F depends nontrivially on  (x), the equation is of nth order.

If a differential equation is satisfied by a function which identically vanishes (i.e. f (x)=0 for each x in the domain of interest), then the equation is said to be homogeneous. Otherwise it is said to be nonhomogeneous (or inhomogeneous). Many differential equations can be expressed in the form where L is a differential operator (with g(x)=0 for the homogeneous case). If the operator L is linear in f, then the equation is said to be a linear ODE and otherwise nonlinear.

Other types of differential equations involve more complicated relations involving the unknown function. A partial differential equation (PDE) is a differential equation where the unknown function depends on more than one variable. In a delay differential equation (DDE), the unknown function depends on the state of the system at some instant in the past.

Solving differential equations is a difficult task. Three major types of approaches are possible:

• Exact methods are generally restricted to equations of low order and/or to linear systems.

• Qualitative methods do not give explicit formula for the solutions, but provide information pertaining to the asymptotic behavior of the system.

• Finally, numerical methods allow constructing approximated solutions.

Liouville formula.

Theorem (The Liouville formula) Let be a sequence of solutions of is continuous. Then the Wronskian of this sequence satisfies the identity

for all .

Recall that the trace trace A of the matrix A is the sum of all the diagonal entries of the matrix.

Proof. Let the entries of the matrix ( then

We use the following formula for differentiation of the determinant, which follows from the full expansion of the determinant and the product rule:

Indeed, if are real-valued differentiable functions then the product rule implies by induction

Hence, when differentiating the full expansion of the determinant, each term of the determinant gives rise to n terms where one of the multiples is replaced by its derivative. Combining properly all such terms, we obtain the derivative of the determinant is the sum of n determinants where one of the rows is replaced by its derivative, that is, (1).

The fact that each vector satisfies the equation can be written in the coordinate form as follows