Applications to Algebraical Series.
13.Summation of Series.—If ur, denotes the (r + 1)th term of a series, and if vris a function of r such that Δvr= urfor all integral values of r, then the sum of the terms um, um+1, ... unis vn+1− vm. Thus the sum of a number of terms of a series may often be found by inspection, in the same kind of way that an integral is found.
14.Rational Integral Functions.—(i.) If uris a rational integral function of r of degree p, then Δur, is a rational integral function of r of degree p − 1.
(ii.) A particular case is that of afactorial,i.e.a product of the form (r + a + 1) (r + a + 2) ... (r + b), each factor exceeding the preceding factor by 1. We have
Δ · (r + a + 1) (r + a + 2) ... (r + b) = (b − a)·(r + a + 2) ... (r + b),
whence, changing a into a-1,
Σ(r + a + 1) (r + a + 2) ... (r + b) =const.+ (r + a)(r + a + 1) ... (r + b)/(b − a + 1).
A similar method can be applied to the series whose (r + 1)th term is of the form 1/(r + a + 1) (r + a + 2) ... (r + b).
(iii.) Any rational integral function can be converted into the sum of a number of factorials; and thus the sum of a series of which such a function is the general term can be found. For example, it may be shown in this way that the sum of the pth powers of the first n natural numbers is a rational integral function of n of degree p + 1, the coefficient of np+1being 1/(p + 1).
15.Difference-equations.—The summation of the series ... + un+2+ un-1+ unis a solution of thedifference-equationΔvn= un+1, which may also be written (E − 1)vn= un+1. This is a simple form of difference-equation. There are several forms which have been investigated; a simple form, more general than the above, is thelinear equationwithconstant coefficients—
vn+m+ a1vn+m-1+ a2vn+m-2+ ... + amvn= N,
where a1, a2, ... amare constants, and N is a given function of n. This may be written
(Em+ a1Em-1+ ... + am)vn= N
or
(E − p1)(E − p2) ... (E − pm)vn= N.
The solution, if p1, p2, ... pmare all different, is vn= C1p1n+ C2p2n+ ... + Cmpmn+ Vn, where C1, C2... are constants, and vn= Vnis any one solution of the equation. The method of finding a value for Vndepends on the form of N. Certain modifications are required when two or more of the p’s are equal.
It should be observed, in all cases of this kind, that, in describing C1, C2as “constants,” it is meant that the value of any one, as C1, is the same for all values of n occurring in the series. A “constant” may, however, be a periodic function of n.
Applications to Continuous Functions.
16. The cases of greatest practical importance are those in which u is a continuous function of x. The terms u1, u2... of the series then represent the successive values of u corresponding to x = x1, x2.... The important applications of the theory in these cases are to (i.) relations between differences and differential coefficients, (ii.)interpolation, or the determination of intermediate values of u, and (iii.) relations between sums and integrals.
17. Starting from any pair of values x0and u0, we may suppose the interval h from x0to x1to be divided into q equal portions. If we suppose the corresponding values of u to be obtained, and their differences taken, the successive advancing differences of u0being denoted by ∂u0, ∂²u0..., we have (§ 3 (ii.))
When q is made indefinitely great, this (writing ƒ(x) for u) becomes Taylor’s Theorem (Infinitesimal Calculus)
which, expressed in terms of operators, is
This gives the relation between Δ and D. Also we have
and, if p is any integer,
From these equations up/qcould be expressed in terms of u0, u1, u2, ...; this is a particular case of interpolation (q.v.).
18.Differences and Differential Coefficients.—The various formulae are most quickly obtained by symbolical methods;i.e.by dealing with the operators Δ, E, D, ... as if they were algebraical quantities. Thus the relation E = ehD(§ 17) gives
hD = loge(1 + Δ) = Δ − ½Δ² +1⁄3Δ³ ...
or
h(du/dx)0= Δu0− ½Δ²u0+1⁄3Δ³u0....
The formulae connecting central differences with differential coefficients are based on the relations μ = cosh ½hD = ½(e1/2hD+ e-1/2hD), δ = 2 sinh ½hD − e1/2hD− e-1/2hD, and may be grouped as follows:—
When u is a rational integral function of x, each of the above series is a terminating series. In other cases the series will be an infinite one, and may be divergent; but it may be used for purposes of approximation up to a certain point, and there will be a “remainder,” the limits of whose magnitude will be determinate.
19.Sums and Integrals.—The relation between a sum and an integral is usually expressed by theEuler-Maclaurin formula. The principle of this formula is that, if umand um+1, are ordinates of a curve, distant h from one another, then for a first approximation to the area of the curve between umand um+1we have ½h(um+ um+1), and the difference between this and the true value of the area can be expressed as the difference of two expressions, one of which is a function of xm, and the other is the same function of xm+1. Denoting these by φ(xm) and φ(xm+1), we have
Adding a series of similar expressions, we find
The function φ(x) can be expressed in terms either of differential coefficients of u or of advancing or central differences; thus there are three formulae.
(i.) The Euler-Maclaurin formula, properly so called, (due independently to Euler and Maclaurin) is
where B1, B2, B3... areBernoulli’s numbers.
(ii.) If we express differential coefficients in terms of advancing differences, we get a theorem which is due to Laplace:—
For practical calculations this may more conveniently be written
where accented differences denote that the values of u are read backwards from un;i.e.Δ′undenotes un-1− un, not (as in § 10) un− un-1.
(iii.) Expressed in terms of central differences this becomes
(iv.) There are variants of these formulae, due to taking hum+1/2as the first approximation to the area of the curve between umand um+1; the formulae involve the sum u1/2+ u3/2+ ... + un-1/2≡ σ(un− u0) (seeMensuration).
20. The formulae in the last section can be obtained by symbolical methods from the relation
Thus for central differences, if we write θ ≡ ½hD, we have μ = cosh θ, δ = 2 sinh θ, σ = δ-1, and the result in (iii.) corresponds to the formula
sinh θ = θ cosh θ/(1 +1⁄3sinh² θ −2⁄3·5sinh4θ +2·4⁄3·5·7sinh6θ − ...).
References.—There is no recent English work on the theory of finite differences as a whole. G. Boole’sFinite Differences(1st ed., 1860, 2nd ed., edited by J. F. Moulton, 1872) is a comprehensive treatise, in which symbolical methods are employed very early. A. A. Markoff’sDifferenzenrechnung(German trans., 1896) contains general formulae. (Both these works ignore central differences.)Encycl. der math. Wiss.vol. i. pt. 2, pp. 919-935, may also be consulted. An elementary treatment of the subject will be found in many text-books,e.g.G. Chrystal’sAlgebra(pt. 2, ch. xxxi.). A. W. Sunderland,Notes on Finite Differences(1885), is intended for actuarial students. Various central-difference formulae with references are given inProc. Lond. Math. Soc.xxxi. pp. 449-488. For other references seeInterpolation.
References.—There is no recent English work on the theory of finite differences as a whole. G. Boole’sFinite Differences(1st ed., 1860, 2nd ed., edited by J. F. Moulton, 1872) is a comprehensive treatise, in which symbolical methods are employed very early. A. A. Markoff’sDifferenzenrechnung(German trans., 1896) contains general formulae. (Both these works ignore central differences.)Encycl. der math. Wiss.vol. i. pt. 2, pp. 919-935, may also be consulted. An elementary treatment of the subject will be found in many text-books,e.g.G. Chrystal’sAlgebra(pt. 2, ch. xxxi.). A. W. Sunderland,Notes on Finite Differences(1885), is intended for actuarial students. Various central-difference formulae with references are given inProc. Lond. Math. Soc.xxxi. pp. 449-488. For other references seeInterpolation.
(W. F. Sh.)
DIFFERENTIAL EQUATION,in mathematics, a relation between one or more functions and their differential coefficients. The subject is treated here in two parts: (1) an elementary introduction dealing with the more commonly recognized types of differential equations which can be solved by rule; and (2) the general theory.
Part I.—Elementary Introduction.
Of equations involving only one independent variable, x (known asordinarydifferential equations), and one dependent variable, y, and containing only the first differential coefficient dy/dx (and therefore said to be of the firstorder), the simplest form is that reducible to the type
dy/dx = ƒ(x)/F(y),
leading to the result ƒF(y)dy − ƒƒ(x)dx = A, where A is an arbitrary constant; this result is said to solve the differential equation, the problem of evaluating the integrals belonging to the integral calculus.
Another simple form is
dy/dx + yP = Q,
where P, Q are functions of x only; this is known as the linear equation, since it contains y and dy/dx only to the first degree. If ƒPdx = u, we clearly have
so that y = e-u(ƒeuQdx + A) solves the equation, and is the only possible solution, A being an arbitrary constant. The rule for the solution of the linear equation is thus to multiply the equation by eu, where u = ƒPdx.
A third simple and important form is that denoted by
y = px + ƒ(p),
where p is an abbreviation for dy/dx; this is known as Clairaut’s form. By differentiation in regard to x it gives
where
thus, either (i.) dp/dx = 0, that is, p is constant on the curve satisfying the differential equation, which curve is thus any one of the straight lines y = cx = ƒ(c), where c is an arbitrary constant, or else, (ii.) x + ƒ′(p) = 0; if this latter hypothesis be taken, and p be eliminated between x + ƒ′(p) = 0 and y = px + ƒ(p), a relation connecting x and y, not containing an arbitrary constant, will be found, which obviously represents the envelope of the straight lines y = cx + ƒ(c).
In general if a differential equation φ(x, y, dy/dx) = 0 be satisfied by any one of the curves F(x, y, c) = 0, where c is an arbitrary constant, it is clear that the envelope of these curves, when existent, must also satisfy the differential equation; for this equation prescribes a relation connecting only the co-ordinates x, y and the differential coefficient dy/dx, and these three quantities are the same at any point of the envelope for the envelope and for the particular curve of the family which there touches the envelope. The relation expressing the equation of the envelope is called asingularsolution of the differential equation, meaning anisolatedsolution, as not being one of a family of curves depending upon an arbitrary parameter.
An extended form of Clairaut’s equation expressed by
y = xF(p) + ƒ(p)
may be similarly solved by first differentiating in regard to p, when it reduces to a linear equation of which x is the dependent and p the independent variable; from the integral of this linear equation, and the original differential equation, the quantity p is then to be eliminated.
Other types of solvable differential equations of the first order are (1)
M dy/dx = N,
where M, N are homogeneous polynomials in x and y, of the same order; by putting v = y/x and eliminating y, the equation becomes of the first type considered above, in v and x. An equation (aB ≷ bA)
(ax + by + c)dy/dx = Ax + By + C
may be reduced to this rule by first putting x + h, y + k for x and y, and determining h, k so that ah + bk + c = 0, Ah + Bk + C = 0.
(2) An equation in which y does not explicitly occur,
ƒ(x, dy/dx) = 0,
may, theoretically, be reduced to the type dy/dx = F(x); similarly an equation F(y, dy/dx) = 0.
(3) An equation
ƒ(dy/dx, x, y) = 0,
which is an integral polynomial in dy/dx, may, theoretically, be solved for dy/dx, as an algebraic equation; to any root dy/dx = F1(x, y) corresponds, suppose, a solution φ1(x, y, c) = 0, where c is an arbitrary constant; the product equation φ1(x, y, c)φ2(x, y, c) ... = 0, consisting of as many factors as there were values of dy/dx, is effectively as general as if we wrote φ1(x, y, c1)φ2(x, y, c2) ... = 0; for, to evaluate the first form, we must necessarily consider the factors separately, and nothing is then gained by the multiple notation for the various arbitrary constants. The equation φ1(x, y, c)φ2(x, y, c) ... = 0 is thus the solution of the given differential equation.
In all these cases there is, except for cases of singular solutions, one and only one arbitrary constant in the most general solution of the differential equation; that this must necessarily be so we may take as obvious, the differential equation being supposed to arise by elimination of this constant from the equation expressing its solution and the equation obtainable from this by differentiation in regard to x.
A further type of differential equation of the first order, of the form
dy/dx = A + By + Cy²
in which A, B, C are functions of x, will be briefly considered below under differential equations of the second order.
When we pass to ordinary differential equations of the second order, that is, those expressing a relation between x, y, dy/dx and d²y/dx², the number of types for which the solution can be found by a known procedure is very considerably reduced. Consider the general linear equation
where P, Q, R are functions of x only. There is no method always effective; the main general result for such a linear equation is that if any particular function of x, say y1, can be discovered, for which
then the substitution y = y1η in the original equation, with R on the right side, reduces this to a linear equation of the first order with the dependent variable dη/dx. In fact, if y = y1η we have
and thus
if then
and z denote dη/dx, the original differential equation becomes
From this equation z can be found by the rule given above for the linear equation of the first order, and will involve one arbitrary constant; thence y = y1η = y1∫ zdx + Ay1, where A is another arbitrary constant, will be the general solution of the original equation, and, as was to be expected, involves two arbitrary constants.
The case of most frequent occurrence is that in which the coefficients P, Q are constants; we consider this case in some detail. If θ be a root of the quadratic equation θ² + θP + Q = 0, it can be at once seen that a particular integral of the differential equation with zero on the right side is y1= eθx. Supposing first the roots of the quadratic equation to be different, and φ to be the other root, so that φ + θ = -P, the auxiliary differential equation for z, referred to above, becomes dz/dx + (θ − φ)z = Re-θxwhich leads to ze(θ-φ)= B + ∫ Re-θxdx, where B is an arbitrary constant, and hence to
y = Aeθx+ eθx∫Be(φ-θ)xdx + eθx∫e(φ-θ)x∫Re-θxdxdx,
or say to y = Aeθx+ Ceθx+ U, where A, C are arbitrary constants and U is a function of x, not present at all when R = 0. If the quadratic equation θ² + Pθ + Q = 0 has equal roots, so that 2θ = -P, the auxiliary equation in z becomes dz/dx = Reθxgiving z = B + ∫ Reθxdx, where B is an arbitrary constant, and hence
y = (A + Bx)eθx+ eθx∫ ∫Re-θxdxdx,
or, say, y = (A + Bx)eθx+ U, where A, B are arbitrary constants, and U is a function of x not present at all when R = 0. The portion Aeθx+ Beθxor (A + Bx)eθxof the solution, which is known as thecomplementary function, can clearly be written down at once by inspection of the given differential equation. The remaining portion U may, by taking the constants in the complementary function properly, be replaced by any particular solution whatever of the differential equation
for if u be any particular solution, this has a form
u = A0eθx+ B0eφx+ U,
or a form
u = (A0+ B0x) eθx+ U;
thus the general solution can be written
(A − A0)eθx+ (B − B0)eθx+ u, or {A − A0+ (B − B0)x} eθx+ u,
where A − A0, B − B0, like A, B, are arbitrary constants.
A similar result holds for a linear differential equation of any order, say
where P1, P2, ... Pnare constants, and R is a function of x. If we form the algebraic equation θn+ P1θn-1+ ... + Pn= 0, and all the roots of this equation be different, say they are θ1, θ2, ... θn, the general solution of the differential equation is
y = A1eθ1 x+ A2eθ2 x+ ... + Aneθn x+ u,
where A1, A2, ... Anare arbitrary constants, and u is anyparticular solution whatever; but if there be one root θ1repeated r times, the terms A1eθ1x+ ... + Areθrxmust be replaced by (A1+ A2x + ... + Arxr-1)eθ1xwhere A1, ... Anare arbitrary constants; the remaining terms in the complementary function will similarly need alteration of form if there be other repeated roots.
To complete the solution of the differential equation we need some method of determining a particular integral u; we explain a procedure which is effective for this purpose in the cases in which R is a sum of terms of the form eaxφ(x), where φ(x) is an integral polynomial in x; this includes cases in which R contains terms of the form cos bx·φ(x) or sin bx·φ(x). Denote d/dx by D; it is clear that if u be any function of x, D(eaxu) = eaxDu + aeaxu, or say, D(eaxu) = eax(D + a)u; hence D²(eaxu),i.e.d²/dx² (eaxu), being equal to D(eaxv), where v = (D + a)u, is equal to eax(D + a)v, that is to eax(D + a)²u. In this way we find Dn(eaxu) = eax(D + a)nu, where n is any positive integer. Hence if ψ(D) be any polynomial in D with constant coefficients, ψ(D) (eaxu) = eaxψ(D + a)u. Next, denoting ∫ udx by D-1u, and any solution of the differential equation dz/dx + az = u by z = (d + a)-1u, we have D[eax(D + a)-1u] = D(eaxz) = eax(D + a)z = eaxu, so that we may write D-1(eaxu) = eax(D + a)-1u, where the meaning is that one value of the left side is equal to one value of the right side; from this, the expression D-2(eaxu), which means D-1[D-1(eaxu)], is equal to D-1(eaxz) and hence to eax(D + a)-1z, which we write eax(D + a)-2u; proceeding thus we obtain
D-n(eaxu) = eax(D + a)-nu,
where n is any positive integer, and the meaning, as before, is that one value of the first expression is equal to one value of the second. More generally, if ψ(D) be any polynomial in D with constant coefficients, and we agree to denote by [1/ψ(D)]u any solution z of the differential equation ψ(D)z = u, we have, if v = [1/ψ(D + a)]u, the identity ψ(D)(eaxv) = eaxψ(D + a)v = eaxu, which we write in the form
This gives us the first step in the method we are explaining, namely that a solution of the differential equation ψ(D)y = eaxu + ebxv + ... where u, v, ... are any functions of x, is any function denoted by the expression
It is now to be shown how to obtain one value of [1/ψ(D + a)]u, when u is a polynomial in x, namely one solution of the differential equation ψ(D + a)z = u. Let the highest power of x entering in u be xm; if t were a variable quantity, the rational fraction in t, 1/ψ(t + a) by first writing it as a sum of partial fractions, or otherwise, could be identically written in the form
Krt-r+ Kr-1t-r+1+ ... + K1t-1+ H + H1t + ... + Hmtm+ tm+1φ(t)/ψ(t + a),
where φ(t) is a polynomial in t; this shows that there exists an identity of the form
1 = ψ(t + a)(Krt−r+ ... + K1t−1+ H + H1t + ... + Hmtm) + φ(t)tm+1,
and hence an identity
u = ψ(D + a) [KrD−r+ ... + K1D−1+ H + H1D + ... + HmDm] u + φ(D) Dm+1u;
in this, since u contains no power of x higher than xm, the second term on the right may be omitted. We thus reach the conclusion that a solution of the differential equation ψ(D + a)z = u is given by
z = (KrD−r+ ... + K1D−1+ H + H1D + ... + HmDm)u,
of which the operator on the right is obtained simply by expanding 1/ψ(D + a) in ascending powers of D, as if D were a numerical quantity, the expansion being carried as far as the highest power of D which, operating upon u, does not give zero. In this form every term in z is capable of immediate calculation.
Example.—For the equation
the roots of the associated algebraic equation (θ² + 1)² = 0 are θ = ±i, each repeated; the complementary function is thus
(A + Bx)eix+ (C + Dx)e−ix,
where A, B, C, D are arbitrary constants; this is the same as
(H + Kx) cos x + (M + Nx) sin x,
where H, K, M, N are arbitrary constants. To obtain a particular integral we must find a value of (1 + D²)−²x³ cos x; this is the real part of (1 + D²)−² eixx³ and hence of eix[1 + (D + i)²]−² x³
or
eix[2iD(1 + ½iD)]−² x³,
or
−¼eixD−² (1 + iD − ¾D² − ½iD³ +5⁄16D4+3⁄16iD5...)x³,
or
−¼eix(1⁄20x5+ ¼ix4− ¾x³ −3⁄2ix² +15⁄8x +9⁄8i);
the real part of this is
−¼ (1⁄20x5− ¾x² +15⁄8x) cos x + ¼ (¼x4−3⁄4x² +9⁄8) sin x.
This expression added to the complementary function found above gives the complete integral; and no generality is lost by omitting from the particular integral the terms −15⁄32x cos x +9⁄32sin x, which are of the types of terms already occurring in the complementary function.
The symbolical method which has been explained has wider applications than that to which we have, for simplicity of explanation, restricted it. For example, if ψ(x) be any function of x, and a1, a2, ... anbe different constants, and [(t + a1) (t + a2) ... (t + an)]−1when expressed in partial fractions be written Σcm(t + am)−1, a particular integral of the differential equation (D + a1)(D + a2) ... (D + an)y = ψ(x) is given by
y = Σcm(D + am)−1ψ(x) = Σcm(D + am)−1e−amxeamxψ(x) = Σcme−amxD−1(eamxψ(x))= Σcme−amx∫eamxψ(x)dx.
The particular integral is thus expressed as a sum of n integrals. A linear differential equation of which the left side has the form
where P1, ... Pnare constants, can be reduced to the case considered above. Writing x = etwe have the identity
When the linear differential equation, which we take to be of the second order, has variable coefficients, though there is no general rule for obtaining a solution in finite terms, there are some results which it is of advantage to have in mind. We have seen that if one solution of the equation obtained by putting the right side zero, say y1, be known, the equation can be solved. If y2be another solution of
there being no relation of the form my1+ ny2= k, where m, n, k are constants, it is easy to see that
so that we have
y1′y2− y1y2′ = A exp.(∫Pdx),
where A is a suitably chosen constant, and exp. z denotes ez. In terms of the two solutions y1, y2of the differential equation having zero on the right side, the general solution of the equation with R = φ(x) on the right side can at once be verified to be Ay1+ By2+ y1u − y2v, where u, v respectively denote the integrals
u =∫y2φ(x) (y1′y2− y2′y1)−1dx, v =∫y1φ(x) (y1′y2− y2′y1)−1dx.
The equation
by writing y = v exp. (-½ ∫ Pdx), is at once seen to be reduced to d²v/dx² + Iv = 0, where I = Q − ½dP/dx − ¼P². If η = − 1/v dv/dx, the equation d²v/dx² + Iv = 0 becomes dη/dx = I + η², a non-linear equation of the first order.
More generally the equation
where A, B, C are functions of x, is, by the substitution
reduced to the linear equation
The equation
known as Riccati’s equation, is transformed into an equation of the same form by a substitution of the form η = (aY + b)/(cY + d), where a, b, c, d are any functions of x, and this fact may be utilized to obtain a solution when A, B, C have special forms; in particular if any particular solution of the equation be known, say η0, thesubstitution η = η0− 1/Y enables us at once to obtain the general solution; for instance, when
a particular solution is η0= √(-A/C). This is a case of the remark, often useful in practice, that the linear equation
where μ is a constant, is reducible to a standard form by taking a new independent variable z = ∫ dx[φ(x)]-½.
We pass to other types of equations of which the solution can be obtained by rule. We may have cases in which there are two dependent variables, x and y, and one independent variable t, the differential coefficients dx/dt, dy/dt being given as functions of x, y and t. Of such equations a simple case is expressed by the pair
wherein the coefficients a, b, c, a′, b′, c′, are constants. To integrate these, form with the constant λ the differential coefficient of z = x + λy, that is dz/dt = (a + λa′)x + (b + λb′)y + c + λc′, the quantity λ being so chosen that b + λb′ = λ(a + λa′), so that we have dz/dt = (a + λa′)z + c + λc′; this last equation is at once integrable in the form z(a + λa′) + c + λc′ = Ae(a + λa′)t, where A is an arbitrary constant. In general, the condition b + λb′ = λ(a + λa′) is satisfied by two different values of λ, say λ1, λ2; the solutions corresponding to these give the values of x +λ1y and x + λ2y, from which x and y can be found as functions of t, involving two arbitrary constants. If, however, the two roots of the quadratic equation for λ are equal, that is, if (a − b′)² + 4a′b = 0, the method described gives only one equation, expressing x + λy in terms of t; by means of this equation y can be eliminated from dx/dt = ax + by + c, leading to an equation of the form dx/dt = Px + Q + Re(a + λa′)t, where P, Q, R are constants. The integration of this gives x, and thence y can be found.
A similar process is applicable when we have three or more dependent variables whose differential coefficients in regard to the single independent variables are given as linear functions of the dependent variables with constant coefficients.
Another method of solution of the equations
dx/dt = ax + by + c, dy/dt = a′x + b′y + c′,
consists in differentiating the first equation, thereby obtaining
from the two given equations, by elimination of y, we can express dy/dt as a linear function of x and dx/dt; we can thus form an equation of the shape d²x/dt² = P + Qx + Rdx/dt, where P, Q, R are constants; this can be integrated by methods previously explained, and the integral, involving two arbitrary constants, gives, by the equation dx/dt = ax + by + c, the corresponding value of y. Conversely it should be noticed that any single linear differential equation
where u, v, w are functions of t, by writing y for dx/dt, is equivalent with the two equations dx/dt = y, dy/dt = u + vx + wy. In fact a similar reduction is possible for any system of differential equations with one independent variable.
Equations occur to be integrated of the form
Xdx + Ydy + Zdz = 0,
where X, Y, Z are functions of x, y, z. We consider only the case in which there exists an equation φ(x, y, z) = C whose differential
is equivalent with the given differential equation; that is, μ being a proper function of x, y, z, we assume that there exist equations
these equations require
and hence
conversely it can be proved that this is sufficient in order that μ may exist to render μ(Xdx + Ydy + Zdz) a perfect differential; in particular it may be satisfied in virtue of the three equations such as
in which case we may take μ = 1. Assuming the condition in its general form, take in the given differential equation a plane section of the surface φ = C parallel to the plane z, viz. put z constant, and consider the resulting differential equation in the two variables x, y, namely Xdx + Ydy = 0; let ψ(x, y, z) = constant, be its integral, the constant z entering, as a rule, in ψ because it enters in X and Y. Now differentiate the relation ψ(x, y, z) = ƒ(z), where ƒ is a function to be determined, so obtaining
there exists a function σ of x, y, z such that
because ψ = constant, is the integral of Xdx + Ydy = 0; we desire to prove that ƒ can be chosen so that also, in virtue of ψ(x, y, z) = ƒ(z), we have
if this can be proved the relation ψ(x, y, z) − ƒ(z) = constant, will be the integral of the given differential equation. To prove this it is enough to show that, in virtue of ψ(x, y, z) = ƒ(z), the function ∂ψ/∂x − σZ can be expressed in terms of z only. Now in consequence of the originally assumed relations,
we have
and hence
this shows that, as functions of x and y, ψ is a function of φ (see the note at the end of part i. of this article, on Jacobian determinants), so that we may write ψ = F(z, φ), from which
in virtue of ψ(x, y, z) = ƒ(z), and ψ = F(z, φ), the function φ can be written in terms of z only, thus ∂F/∂z can be written in terms of z only, and what we required to prove is proved.
Consider lastly a simple type of differential equation containingtwoindependent variables, say x and y, and one dependent variable z, namely the equation
where P, Q, R are functions of x, y, z. This is known as Lagrange’s linear partial differential equation of the first order. To integrate this, consider first the ordinary differential equations dx/dz = P/R, dy/dz = Q/R, and suppose that two functions u, v, of x, y, z can be determined, independent of one another, such that the equations u = a, v = b, where a, b are arbitrary constants, lead to these ordinary differential equations, namely such that
Then if F(x, y, z) = 0 be a relation satisfying the original differential equations, this relation giving rise to
It follows that the determinant of three rows and columns vanishes whose first row consists of the three quantities ∂F/∂x, ∂F/∂y, ∂F/∂z, whose second row consists of the three quantities ∂u/∂x, ∂u/∂y, ∂u/∂z, whose third row consists similarly of the partial derivatives of v. The vanishing of this so-called Jacobian determinant is known to imply that F is expressible as a function of u and v, unless these are themselves functionally related, which is contrary to hypothesis (see the note below on Jacobian determinants). Conversely, any relation φ(u, v) = 0 can easily be proved, in virtue of the equations satisfied by u and v, to lead to