Differential Equations

Higher Order Equations

Nth Order Diff Equations Logo

Equations Solvable in Quadratures

It is said that the differential equation is solved in quadratures if its general solution is expressed in terms of one or more integrals.

Next, we consider three types of higher-order equations that are integrated in quadratures.

Case Equation of Type

Assume first that this equation can be transformed into an explicit form for the derivative i.e. expressed as

We integrate this equation times consecutively in the range from to As a result, we obtain the following expressions for the derivatives and the function

The last formula is the general solution of differential equation in quadratures.

At we obtain the particular solution satisfying the initial conditions

where is a given set of numbers.

The iterated integral in the expression for can be converted to a single integral. Indeed, in the case we consider the integral

where denotes a variable of integration in the inner integral.

This iterated integral is defined in the triangular region shown in Figure

The iterated integral is defined in the triangular region D
Figure 1.

We can change the order of integration in this integral, using Dirichlet's formula:

As a result, the double integral reduces to a single integral:

Similarly, we can simplify the the triple iterated integral in the case

For the iterated integral of arbitrary multiplicity the following expression is valid:

which is called the Cauchy formula for iterated integrals.

The resulting expression is a particular solution of the differential equation with zero initial conditions:

Accordingly, the general solution of the original equation is described by

Note that the Cauchy formula relates the function and its th order derivative If we assume that can be a real number, then we arrive at the concept of fractional order derivative.

Instead of the factorial in the Cauchy's formula we can write the so-called gamma function which is continuous and expressed through the improper integral in the form

A schematic view of the gamma function for real values of is shown in Figure

A schematic view of the gamma function
Figure 2.

For natural values of the following equality holds:

Then the Cauchy formula is represented as follows:

where is a real number.

This formula can be considered as the definition of fractional derivative of order if the original function is known, or as the definition of the integral or fractional order if the corresponding derivative is given.

We have considered the solution of the explicit differential equation in quadratures. The implicit equation can be also integrated if it can be solved with respect to or more generally, presented in parametric form:

Then, given that

we have

Similarly, we find the other derivatives and the function As a result, we obtain the general solution of the equation in parametric form:

Case Equation of Type

Consider first the case when such an equation can be solved for

We solve it as follows. We introduce the new variable Then the equation can be written as

Separating the variables, we find its general solution:

Returning to the variable we obtain the differential equation of the th order:

which is solved by the method set out in paragraph above.

The general implicit equation can be integrated if it is represented in parametric form as

As we obtain the following expression for

The expression for is found by successive integration:

As a result, we obtain the general solution in parametric form.

Case Equation of Type

Suppose that the equation is solved for

Introducing the new variable we write it as

Multiplying both sides by (under the assumption that the equation has no solution ), we obtain:

It is evident that we have an equation of the form which was considered in paragraph and which can be solved in quadratures.

If the equation has a solution then the general solution is given by

In the case where the differential equation admits a parametric representation

its solution is constructed as follows. It follows from the relationships

that

or in parametric form:

Integrating, we find:

Now we know the parametric expression for the derivatives and that is, the problem reduces to type

See solved problems on Page 2.

Page 1 Page 2