Exact Equations of the First Order and of the First Degree

An ordinary differential equation of the first order and of the first degree may be expressed in the form of a total differential equation,

where and are functions of and and do not involve . If the differential is immediately, that is without multiplication by any factor, expressible in the form , where is a function of and , it is said to be exact.

If the equation

is exact and its primitive is¹

the two expressions for , namely,

must be identical, that is,

Then

provided that the equivalent expression is continuous. The condition of integrability (A) is therefore necessary. It remains to show that the condition is sufficient, that is to say, if it is satisfied the equation is exact and its primitive can be found by a quadrature.

Let be defined by

where is an arbitrary constant, and is a function of alone which, for the moment, is also arbitrary. Then will be a primitive of

if

The first condition is satisfied; the second determines thus:

and therefore

where is arbitrary.

The condition is therefore sufficient, for the equation is exact and has the primitive

The constants and may be chosen as is convenient, there are not, in all, three arbitrary constants but only one, for a change in or in is equivalent to adding a constant to the left-hand member of the primitive. This is obvious as far as is concerned, and as regards , it is a consequence of the condition of integrability.

1. Separation of Variables

A particular instance of an exact equation occurs when is a function of alone and a function of alone. In this case may be written for and for . The equation

is then said to have separated variables. Its primitive is

When the equation is such that can be factorised into a function of alone and a function of alone, and can similarly be factorised into and , the variables are said to be separable, for the equation

may be written in the separated form

It must be noticed, however, that a number of solutions are lost in the division of the equation by . If, for example, is a root of the equation , it would furnish a solution of the equation (I) but not necessarily of the equation (II).

2. Homogeneous Equations

If and are homogeneous functions of and of the same degree , the equation is reducible by the substitution² to one whose variables are separable. For

and therefore

becomes

or

where

The solution is

When the equation

is both homogeneous and exact, it is immediately integrable without the introduction of a quadrature, provided that its degree of homogeneity is not -1. Its primitive is, in fact,

For let , then

by Euler's theorem (1.2.3.2), and similarly

Consequently

and therefore

Hence if , the primitive is

When the integration in general involves a quadrature. It is a noteworthy fact that the homogeneous equation

is exact, for the condition of integrability, namely

reduces to

which is true, by Euler's theorem, since and are homogeneous and of the same degree. Thus any homogeneous equation may be made exact by introducing the integrating factor . The degree of homogeneity of this exact equation is, however, -1, so that the integration of a homogeneous equation in general involves a quadrature.

An equation of the type

in which are constants such that , may be brought into the homogeneous form by a linear transformation of the variables, for let

where are new variables and are constants such that

The equation becomes

so that is a homogeneous function of of degree zero. The constants are determinate since .

When , let be a new dependent variable defined by

then

The variables are now separable.

3. Linear Equations of the First Order

The most general linear equation of the first order is of the type

where and are functions of alone. Consider first of all the homogeneous linear equation³

Its variables are separable, thus:

and the solution is

where is a constant.
Now substitute in the non-homogeneous equation, the expression

in which , a function of , has replaced the constant . The equation becomes

whence

The solution of the general linear equation is therefore

and involves two quadratures.

The method here adopted of finding the solution of an equation by regarding the parameter, or constant of integration of the solution of a simpler equation, as variable, and so determining it that the more general equation is satisfied, is a particular case of what is known as the method of variation of parameters.⁴

It is to be noted that the general solution of the linear equation is linearly dependent upon the constant of integration . Conversely the differential equation obtained by eliminating between any equation

and the derived equation

is linear.

If any particular solution of the linear equation is known, the general solution may be obtained by one quadrature. For let be a solution, then the relation

is satisfied identically. By means of this relation, can be eliminated from the given equation, which becomes

The equation is now homogeneous in , and has the solution

where is the constant of integration.

If two distinct particular solutions are known, the general solution may be expressed directly in terms of them. For it is known that the general solution has the form

and any two particular solutions and are obtained by assigning definite values and to the arbitrary constant , thus

and therefore

4. The Equations of Bernoulli and Jacobi

The equation

in which and are functions of alone, is known as the Bernoulli equation.⁵ It may be brought into the linear form by a change of dependent variable. Let

then

and thus if the given equation is written in the form

it becomes

and is linear in .

The Jacobi equation,⁶

in which the coefficients are constants, is closely connected with the Bernoulli equation. Make the substitution

where are constants to be determined so as to make the coefficients of , and separately homogeneous in and . When this substitution is made, the equation is so arranged that the coefficient of is homogeneous and of the first degree, thus

where

The coefficients of and also become homogeneous if and are so chosen that , or, more symmetrically, if

that is if

">

Thus is determined by the cubic equation

and when is so determined, and are then the solutions of any two of the consistent equations ().

The equation may now be written in the form

The substitution brings it into the form of a Bernoulli equation,

where and are functions of alone.

It will be shown in a later section (§ 2.2.1) that if the three roots of the equation in are and are distinct,⁷ the general solution of the Jacobi equation is

where are linear expressions in and .

5. The Riccati Equation

The equation

in which and are functions of , is known as the generalised Riccati equation.⁸ It is distinguished from the previous equations of this chapter in that it is not, in general, integrable by quadratures. It therefore defines a family of transcendental functions which are essentially distinct from the elementary transcendents.⁹

When any particular solution is known, the general solution may be obtained by means of two successive quadratures. Let

then the equation becomes

and since is a solution, it reduces to

This is a case of the Bernoulli equation; it is reduced to the linear form by the substitution

from which the theorem stated follows immediately.
Let be three distinct particular solutions of the Riccati equation and its general solution. Then

satisfy one and the same linear equation, and consequently

where is a constant. When and are replaced by their expressions in terms of , and this relation may be written

This formula shows that the general solution of the Riccati equation is expressible rationally in terms of any three distinct particular solutions, and also that the anharmonic ratio of any four solutions is constant. It also shows that the general solution is a rational function of the constant of integration. Conversely any function of the type

where are given functions of and an arbitrary constant, satisfies a Riccati equation, as may easily be proved by eliminating between the expressions for and the derived expression for .

When is identically zero, the Riccati equation reduces to the linear equation; when is not zero, the equation may be transformed into a linear equation of the second order. Let be a new dependent variable defined by

then the equation becomes

where

The substitution

now brings the equation into the proposed form, namely,

In particular, the original equation of Riccati, namely,

where and are constants, becomes¹⁰

6. The Euler Equation

An important type of equation with separated variables is the following: ¹¹

in which

Consider first of all the particular equation

one solution is¹²

but the equation has also the solution

Since, as will be proved in Chapter III., the differential equation has but one distinct solution, the two solutions must be related to one another in a definite way. This relation is expressed by the equation

Now let

then

Let , then

and therefore

Thus the addition formula for the sine-function is established.

In the same way, the differential equation

has the solution

where is the inverse Jacobian elliptic function¹³ defined by

Let

then

A second and equivalent solution may be found as follows. By definition

and therefore

Similarly

from which it follows that

Hence

This equation is immediately integrable; the solution is

or

that is

By putting it is found that , and therefore

This is the addition formula for the Jacobian elliptic function .

The same process of integration may be applied to the general Euler equation.¹⁴ In particular it may be noted that when a linear transformation brings the equation into the form

If is the Weierstrassian elliptic function defined by

and , the general solution of the equation is .

An equivalent general solution is

It may thus be shown that the addition-formula for the -function is

Footnotes

1. Throughout this Chapter the letter or generally denotes a constant of integration. Any other use of these letters will be evident from the context. ↩

2. This device was first used by Leibniz in 1691. ↩

3. The term homogeneous is applied to a linear equation when it contains no term independent of and the derivatives of . This usage of the term is to be distinguished from that of the preceding section in which an equation (in general non-linear) was said to be homogeneous when and were homogeneous functions of and of the same degree. There should be no confusion between the two usages of the term. ↩

4. Vide § 5.2.3. The application of the method to the linear equation of the first order is due to John Bernoulli, Acta Erud., 1697, p. 113, but the solution by quadratures was known to Leibniz several years earlier. ↩

5. James Bernoulli, Acta Erud. 1695, p. 553 [Opera 1, p. 663]. The method of solution was discovered by Leibniz, Acta Erud. 1696, p. 145 [Math. Werke 5, p. 329]. ↩

6. J. für Math. 24 (1842), p. 1 [Ges. Werke, 4, p. 256]. See also the Darboux equation, § 2.2.1, infra. ↩

7. The case in which they are not distinct is discussed by Serret, Cale. Diff. et Int. 2, p. 431. ↩

8. Riccati, Acta Erud. Suppl., VIII. (1724), p. 73, investigated the equation , with which his name is usually associated. The generalised equation was studied by d'Alembert, vide infra, § 12.5.1. ↩

9. The elementary transcendents are functions which can be derived from algebraic functions by integration, and the inverses of such functions. Thus the logarithmic function is defined as ; ; its inverse is the exponential function. From the exponential function the trigonometrical and the hyperbolic functions are derived by rational processes, and such functions as the error-function by integration. ↩

10. The solution of this equation may be expressed in terms of Bessel functions (§7.3.1). ↩

11. Euler, Inst. Calc. Int., 1, Chaps. V., VI. ↩

12. The function is defined as. is defined as the inverse of are , so that ; and is defined as with the condition that . No further properties of the trigonometrical functions are assumed. ↩

13. Whittaker and Watson, Modern Analysis, Chap. XXII. ↩

14. Cayley, Elliptic Functions, Chap. XIV. ↩