A Treatise on Infinitesimal Calculus ...

407. A solution being given, to determine whether it is singular

408. Conditions of a singular solution, when the general in-

tegral is found

409. The second mode of satisfying the condition, and the geo-

metrical interpretation of the same

410. The third mode of satisfying the condition

577

578

580

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

..

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

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

422. Determination of functions by means of integration

423. Riccati's equation

601

602

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

603

SECTION 11.-Solution of Geometrical Problems, dependent on

Differential Equations.

426. Trajectories of plane curves referred to rectangular coor-

dinates

427. Trajectories of plane curves referred to polar coordinates

428. Other cases of trajectories..

429. Trajectories of surfaces.

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.

SECTION 1.-General Properties.

433. General remarks

434. Condition that a differential equation of two variables

should be integrable once, independently of the func-

tional relation between its variables ..

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

615

and examples

617

436. An a posteriori proof of the theorem

618

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

622

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

member is dependent on the integration of the same

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 (n-m)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

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

469. Integration of a linear partial differential equation of the

470. Geometrical problems depending on the solution of partial

differential equations of the second order ..

670

CHAP. XVII.

THE SOLUTION OF DIFFERENTIAL EQUATIONS BY THE CALCULUS

OF OPERATIONS.

SECTION 1.-The Solution of Total Differential Equations by

Symbolical Methods.

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-

efficients

683

476. The application to equations which have variable coefficients 685

CHAP. XVIII.

INTEGRATION OF SIMULTANEOUS DIFFERENTIAL EQUATIONS.

477. Simultaneous differential equations of the first order

478. The number of arbitrary constants is the same as that of

687

the simultaneous equations

688

479. Integration of linear simultaneous equations

689

480. Integration of linear simultaneous equations with constant

coefficients and of the first order

481. Linear simultaneous equations of higher orders and of con-

stant coefficients solved by the calculus of operating

symbols

693

482. The same method applied to simultaneous linear partial

differential equations ..

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486. The method of undetermined coefficients ..

487. The solution of Riccati's equation affected by the method
488. A particular case of this equation
489. The general solution of particular forms of Riccati's equa-
tion expressed in terms of a definite integral

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