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
We integrate this equation
The last formula is the general solution of differential equation in quadratures.
At
where
The iterated integral in the expression for
where
This iterated integral is defined in the triangular region
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
which is called the Cauchy formula for iterated integrals.
The resulting expression is a particular solution of the differential equation
Accordingly, the general solution of the original equation is described by
Note that the Cauchy formula relates the function
Instead of the factorial
A schematic view of the gamma function
For natural values of
Then the Cauchy formula is represented as follows:
where
This formula can be considered as the definition of fractional derivative of order
We have considered the solution of the explicit differential equation
Then, given that
we have
Similarly, we find the other derivatives and the function
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
Separating the variables, we find its general solution:
Returning to the variable
which is solved by the method set out in paragraph
The general implicit equation
As
The expression for
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
Multiplying both sides by
It is evident that we have an equation of the form
If the equation
In the case where the differential equation
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