Meaning of Pdx + Qdy + Rdz=0, 272. Condition of derivation from a single primitive, 275. Solution, 276. General rule and ex- amples, 279. Homogeneous equations, 281. Integrating fac- tors, 282. Equations not derivable from a single primitive, 283. More than three variables, 286. Equations of an order higher SIMULTANEOUS DIFFERENTIAL EQUATIONS Meaning of a determinate system, 292. General theory of simulta- neous equations of the first order and degree, 293-307. Systems of two equations, 294. Of more than two, 298. Linear equations with constant coefficients, 300. Equations of an order higher Nature, 319. Primary modes of genesis, 321. Solution when all the differential coefficients have reference to only one of the inde- pendent variables, 322. Linear equations of first order, 324. Their genesis, 325. Their solution, 329. Non-linear equations of the first order, 335. Complete primitive, general primitive, and singular solution, 339. Sufficiency of a single complete pri- mitive, 345. Singular solutions, 346. Geometrical applica- tions, 347. Symmetrical and more general solution of equations PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER 361 The equation Rr+Ss + Tt=V, 361. Condition of its admitting a first integral of the form u=ƒ (v), 362. Deduction of such integral when possible, 364. Relations of first integrals, 366. General rule, 369. Miscellaneous theorems. Poisson's method, 375. Duality, 376. Legendre's Transformation, 379. Exercises, 380. Laws of direct expressions, 381-384. Inverse forms, 385. Linear equations with constant coefficients, 388. Forms purely sym- bolical, 398. Equations solvable by means of the properties of Symbolical form of differential equations with variable coefficients, 412. Finite solution, 415-436. Reduction of binomial equations, 418. Pfaff's equation, 430. Equations not binomial, 432. Solution by series, 437: Evaluation of series, 441. Generalization, 446. Theorem of development, 447. Laplace's reduction of partial differential equations, 450. Miscellaneous notices, 454. Exer- Laplace's method, 461. Partial differential equations, 475. Parseval's theorem, 477. Solution by Fourier's theorem, 478. Miscellaneous The following portions of the work are recommended to beginners. Chap. I. Arts. 1—8, 11. Chap. II. Arts. 1--11. Chap. III. Chap. IV. Arts. 1-9. Chap. V. Arts. 1-4. Chap. VI. Arts. 1-10. Chap. VII. Arts. 1—8. Chap. VIII. Arts. 1–7. Chap. IX. Arts. 1, 3-12. Chap. X. Arts. 1-3. Chap. XI. Arts. 1-7. Chap. XII. Arts. 1-8. Chap. XIII. Chap. XIV. Arts. 1-12. Chap. XV. DIFFERENTIAL EQUATIONS. CHAPTER I. OF THE NATURE AND ORIGIN OF DIFFERENTIAL EQUATIONS. 1. WHAT is meant by a differential equation? To answer this question we must revert to the fundamental conceptions of the Differential Calculus. The Differential Calculus contemplates quantity as subject to variation; and variation as capable of being measured. In comparing any two variable quantities x and y connected by a known relation, e.g. the ordinate and abscissa of a given curve, it defines the rate of variation of the one, y, as referred to that of the other, x, by means of the fundamental conception of a limit; it expresses that ratio by a differential coefficient dy; and of that differential coefficient it shews how da to determine the varying magnitude or value. Or, again, condy sidering as a new variable, it seeks to determine the rate dx of its variation as referred to the same fixed standard, the variation of x, by means of a second differential coefficient day and so on. But in all its applications, as well as in its theory and its processes, the primitive relation between the variables x and y is supposed to be known. In the Integral Calculus, on the other hand, it is the relation among the primitive variables, x, y, &c. which is sought. In that branch of the Integral Calculus with which the student B. D. E. 1 is supposed to be already familiar, the differential coefficient dy dx being given in terms of the independent variable x, it is proposed to determine the most general relation between y and x. Expressing the given relation in the form In (1) we have a particular example of an equation in the expression of which a differential coefficient is involved. But instead of having as in that example expressed in terms of dy dx x, we might have that differential coefficient expressed in terms of y, or in terms of x and y. Or we might have an equation in which differential coefficients of a higher order, d'y d3y dds, &c., were involved, with or without the primitive variables. All these including (1) are examples of differential equations. The essential character consists in the presence of differential coefficients. are seen to be differential equations, the latter of which contains, while the former does not contain, the primitive variables. And thus we are led to the following definition. DEF. A differential equation is an expressed relation involving differential coefficients, with or without the primitive variables from which those differential coefficients are derived. |