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CHAP. XVI.

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SECTION 6.Integration of Partial Differential Equations of

: : Higher Orders. 467. Monge's method .. .. .. .. .. .. .. .. 666 468. Examples in illustration .. .. .. .. .. . . 668 469. Integration of a linear partial differential equation of the

nth order .. .. .. .. .. .. .. .. .. 669 470. 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 Diferential Equations by

Symbolical Methods. 471. The process developed, and the first form of the result .. 673 472. The second form of the result, and examples .. .. ... 677 473. Other modes of employing the operative symbols .. ... 680 474. The process applied to a linear equation whose coefficients

are the successive powers of a binomial.. .... 682

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

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

stant coefficients solved by the calculus of operating symbols .. .. .. .. .. .. .. .. .. 693

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482. The same method applied to simultaneous linear partial

differential equations .. .. .. .. .. .. .. 483. Some particular examples .. .. .. .. .. .. ..

694

CHAP. XIX.

INTEGRATION OF DIFFERENTIAL EQUATIONS BY SERIES.

699

700

701

484. Application of Taylor's theorem
485. Application of Maclaurin's theorem .. .. .. .. ...
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 .. ..

703

704

705

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