The integral of this is easily seen to be x = C1eat + C2 €ßt + C3 €¥1‚ a, B, y being the roots of 3 - (a'b + a′′c + b′′"c′) ≈ + a'b′′c + a′′bc' = 0. The values of y and x are easily derived from that of x. multiplying the second by a and adding, there results αʹ, where h3, k2 are the roots of the equation x2 + (a' + b) ≈ + a'b — ab' = 0. + C1 cos (hx + a) + C2 cos (kx + ß) ; the integral of which is x = cos 2 t 2 sin 2t - cos t + C1 cos (23 t + a) + C2 cos (33 t +ß), and from this the value of y is easily found. Take the system of equations. These may be written under the form (a + a ̧d2 + a ̧d1 + &c.) x + (a1d + a,d3 + &c.) y = m sinnt, (a。 + a2d2 + a ̧d1 + &c.) y − (a ̧d + a ̧d3 + &c.) x = m cos nt, where for convenience the differentials of t are omitted. Eliminating y we have {(a + a‚d2 + a ̧d1 + &c.)2 + (a1d + azd3 + &c.)3} x = m (a + a1n — a ̧n2 — a ̧n3 + a ̧n' + an3 - &c.) sin nt. It is obvious from the form of this that the complementary function must be of the form (A cos λt + B sin λt), where all the values are to be assigned to A which satisfy the equation (ã。 − a¿λ2 + a ̧λ1 + &c.)2 − λ2 (a1 − az λ2 + &c.)2 = 0. Hence we have 0 ∞ = Σ (A cos λt + B sin λt) + - m (a。 + a ̧n — an2 — a‚Ñ3 + a ̧n1 + a ̧n3 — &c.) - The same method is applicable to linear partial differential equations in which the coefficients are constants. The two symbols of differentiation are to be treated as two independent constants, since they do not affect each other, and are both subject to the laws which regulate the combinations of ordinary algebraical symbols. To eliminate u, operate on the first equation with d +b, and on the second with c and subtract: dx dy dx d d 1 and 1 + (cc')3 = —, d n this may ≈ = 0; { + m (a + b) } { ~ + n (a = + } d do dy d.x dy the integral of which is -mbz -nbz %=Є ¤ (y − amx) + e−113 † (y — anx) ; and from this u can be found. (11) The equations for determining the small disturbances of an elastic medium in three dimensions are See Airy's Tracts, p. 279, Note. We might in this case eliminate v and w by a process similar to that used in Ex. (5) of this section; but the following method is more convenient. d 2 Operate on each of these by respectively, and add: then When the equations are not linear there is no general method for integrating them; and therefore the means of doing so must be adapted to the particular case under consideration. Two of the more important examples of such equations, which occur in dynamics, are subjoined. (12) The equations for determining the motion of a particle attracted to a fixed centre of force varying inversely as the square of the distance are Multiply the first equation by y and the second by a, |
