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If the equations M+Nd;y=0, M+Pd,x=0, N+Pd,z=0,(1) may either of them be derived from the same function u, then

d,M=d,N, d. N=d.P, d.N=d,P; ; and un, Ung u, being the integrals obtained from the equations (1) respectively,

u=u, +S:(P-du,),

=u, +S,(N-d,u,),

=u, +S.(M-d.u). (L. D. C. 282–6.) (29.) Separation of the variables. All equations of the form

X + Yd, y=0, in which X and Y are functions of r and y respectively, are immediately integrable ; the integral is

SzX+S,Y=0. The equations Y+X d, y=0, and X,Y+XY,d,y=0 may be reduced to the above form by dividing them by XY. (30.) Homogeneous equations. Any homogeneous equation between two variables, each term of which is of m dimensions may be rendered integrable by dividing every term by *", and assuming y=xx, then

d.y=x+ ad z, and the given equation may be reduced to

2, + Z2(x + xd2x)=1, in which the variables may readily be separated.

The equation a, +bx+c;y+(a, + box + cy)dzy=0 may be rendered homogeneous by assuming a, + b + cy=u, a, + box + ce=v,

67.d,v whence

d.y=

C7.d.V — C2 (31.) The linear equation, dxy+X,y+X,=0.

This equation may be rendered integrable by multiplying every term by ede xı: the required integral is

y=e-/X1{VxX**1 + const.}

(32.) Riccati's equation; d, y + ay® + b20=0.

If m=0, the variables are immediately separable: otherwise

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substituting these values for y and dry respectively, we obtain

wPd2% + ax? + b2cm +4=0. This equation is homogeneous, if m= -2; and if m=-4, the variables are immediately separable: otherwise assume

1

xm + 3

= N12

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and by substituting these values, the equation will be reduced to its original form, and the same process may be repeated.

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THE INTRODUCTION OF A FACTOR WHICH RENDERS A

DIFFERENTIAL EQUATION INTEGRABLE.

(33.) Let M + Ndry=0 be an equation which does not fulfil the condition

d, M=d,N, and let V=c be the primitive equation, and

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then the given equation may be rendered integrable by introducing the factor u. (u): in the simplest case p(u)=1. To determine u we have

u{d,M-d.N}=Ndu- Md,U, this however usually presents greater difficulties than the given equation.

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If u is a function of x only, then d,u=0, and
du

,)=X

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hence

u=flix If u is a function of y only, it may be determined in a similar

manner.

If M and N are homogeneous functions, then

1

U=

Mx + Ny (34.) The required factor may sometimes be conveniently found by transforming the equation

M + Ndry=0 into K + Ldu=0, in which the relations between x, y, t, and u, are known, and K

L finding a factor V such that be a function of t, and

may

V of u: then if the values of t and u, in terms of •x and y, be substituted in V, the quantity obtained will render the given equation integrable. The equation P+Qd,u=0 may be rendered integrable by

1 the factor

if
PT QU

V
P=

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1

V being any function of t and u, and T and U, of t and u, respectively.

The equation X,y + (y + X,)d.y=0 may be rendered integrable by the factor

+ Xy? +{X – a +b(2a – X)*}ay The equation X,y+(x,y + X)d.y=0 may be rendered

y" integrable by the factor

if (1 + Xy)"

X,=bX".d, X+(n-1)a Xn-2.d, X,
X,=(m-n + 2)a X”,
X=bXm+1 + (m + 1)a X"-1.

(L. D. C. 316—23; L. C. D. 567—80.)

SINGULAR SOLUTIONS.

(35.) Let f(x,y,c)=0, c being an arbitrary constant, and let p(x,y,y',c)=0, in which y represents dzy, be derived from the former by differentiation; from these equations, another of the form y(x,y,'')=0 may be obtained by the elimination of c. If u is such a function of x and y, that

P(x,y,y',u)=0, may be derived by differentiation from

f(x,y,u)=0, then y(x,y,y)=0 will result from the elimination of u, and f(x,y,u)=0 may be considered as the primitive equation.

Such values of f(x,yu)=0 as are not included in the general form f(x,y,c)=0 are singular solutions.

The equation du f(x,y,u)=0 must be satisfied by the values of u which give singular solutions,

All singular solutions will be found amongst the equations obtained by substituting in f(x,y,c)=0 those variable values of c which satisfy the equation

d.f(x,y,c)=0. The values of c which give singular solutions satisfy the equations d.ac=co, d,c=; they also render diy=

y=.. If f(x,y,c)=0, the general solution of a differential equation, represents a class of curves, each of which is determined by assigning a particular value to the arbitrary constant c, then the singular solution, which is independent of c, represents the bounding or circumscribing curve.

If a given differential equation f(wy,y)=0 can be solved for y', then a singular solution may be obtained by substituting for y' a value which satisfies the equations dry= 0, d,y= %.

(L. D. C. 324-41.)

For singular solutions of differential equations of higher orders, see L. C. D. 638-58,

(36.) Equations of more than one dimension in y'.

If a constant of n dimensions be eliminated between any equation f(x,y)=0 and its derived equation f'(x,y)=0, an equation may be obtained of the form

y'n + an-14'n – 1 + &c. + any' +a=0: let the roots of this be P P2, &c. and let the equations

y-P=0, Y-P,=0, &c. be integrated separately; then the primitive equation will be the product of the integrals thus obtained.

If we have an equation f(x,y,y)=0, homogeneous with regard to x and y, then by assuming y=uX, we obtain

Suyu

y=UE whence we may frequently obtain an integrable equation between y and u.

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(37.) Clairault's formula : y=y'x + f(y).
By differentiating the given equation we obtain

{*+d, f(y)}d y=0, whence day=0, and .. y=c, by substituting which we have

y=cx + f(c). This is the general solution: a singular solution may be obtained by eliminating y' between the original equation, and

x + dy f(y')=0.

(L. D. C. 342—7; L. C. D. 582—9.)

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