A Treatise on Infinitesimal Calculus ...

SECTION 8.-Differential Equations of first order and of any degree.

577

578
580

411. General method of integration, and examples

412. Particular forms. Clairaut's form

413. Geometrical interpretation of Clairaut's form

414. An extended form

415. Integration of the case wherein one variable can be ex-

pressed explicitly in terms of the other and the derived

function

589

591

416. The case where the coefficients of the powers of y' are

homogeneous

SECTION 9.-Partial Differential Equations of the First Order

and Higher Degrees.

417. The inquiry important, but necessarily imperfect at present 592

418. Charpit's method of solution, and examples

SECTION 10. Various Theorems and Applications.

593

421. The integral of Euler's differential equation

422. Determination of functions by means of integration

423. Riccati's equation

599

601

602

424, 425. Other and equivalent forms of Riccati's equation

SECTION 11.-Solution of Geometrical Problems, dependent on

Differential Equations.

426. Trajectories of plane curves referred to rectangular coor-

430. Geometrical problems involving total differential equations

of three variables

431. Geometrical problems involving partial differential equa-

tions of the first order

611

612

613

432. Integration of the differential equation of the lines of cur-
vature on an ellipsoid

CHAP. XVI.

INTEGRATION OF DIFFERENTIAL EQUATIONS OF ORDERS HIGHER

THAN THE FIRST.

434. Condition that a differential equation of two variables

should be integrable once, independently of the func-

tional relation between its variables ..

615

435. The statement of the converse of the preceding theorem,

437. The conditions that a differential equation should be simi-

larly integrable m times

618

439. Application of the conditions to linear equations

620

SECTION 2.-Linear Differential Equations.

440. Definition of a linear differential equation

441. Proof that the integration of a linear equation with a second

member is dependent on the integration of the same

equation without the second member

442. Application of the process to an equation of the third order
443. Expression of the result in terms of the particular integrals
of the equation without the second member

444. Examples of application of the process
445. If m particular integrals of an equation without the second
member are known, the integration of the equation
with the second member depends on that of an equa-
tion of the (nm)th order

446. The general integral of an equation with the second member

may be expressed in terms of n particular integrals of

the equation without the second member and of a par-

ticular integral of itself

627

628

e

447. A differential equation, linear in at least the first two

terms, may be transformed into another linear equa-

tion of the same order, and without the second term

448. If a relation is given between two particular integrals of a

linear differential equation of the nth order, the order

of the equation may be diminished by unity

449. If n particular integrals of a differential equation, which is

without the second member, are known, the coefficients

of the several terms are functions of these integrals,

and may be found by a process analogous to that of

forming an algebraical equation whose roots are given

SECTION 3.-Linear Differential Equations with Constant

Coefficients.

450. First method of integration

451. The determination of the constants

452. Modification if the roots are impossible

453. Examples of the method ..

454. Modification if two or more roots of the characteristic are

457. The form of the result when the right-hand member of the

equation is a constant

645

458. Lagrange's variation of parameters

646

SECTION 4.-Particular forms of Linear Differential Equations

with Variable Coefficients.

459. Integration of a linear differential equation whose co-

efficients are the successive powers of a binomial

460. Integration of a linear differential equation, whose co-

651

652

654

SECTION 5.-Differential Equations of Higher Orders and Degrees.

462. Integration of ƒ "(x) = F{ƒ"−1(x), ƒ"-2(x)} ..

463. Integration of f(x, y, y") = 0, and of f(y, y', y") = 0

464. Homogeneous equations of the second order

465, 466. Two other particular forms

..

663

664

SECTION 6.-Integration of Partial Differential Equations of

Higher Orders.

467. Monge's method

468. Examples in illustration

....

469. Integration of a linear partial differential equation of the

nth order ..

470. Geometrical problems depending on the solution of partial

differential equations of the second order ..

CHAP. XVII.

THE SOLUTION OF DIFFERENTIAL EQUATIONS BY THE CALCULUS

OF OPERATIONS.

SECTION 1.-The Solution of Total Differential Equations by

471. The process developed, and the first form of the result
472. The second form of the result, and examples
473. Other modes of employing the operative symbols
474. The process applied to a linear equation whose coefficients
are the successive powers of a binomial

SECTION 2.-The Solution of Partial Differential Equations by

Symbolical Methods.

475. The process applied to equations which have constant co-