ay - a'y – b'x = c. Eliminating y by operating on the first with -a', multiplying the second by a and adding, there results Eliminating y we obtain the equation + 6 x = 2 cos 2 t – 4 sin 2t - 2 cost; and from this the value of y is easily found. Take the system of equations. 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 and subtract: + a +b, and on the second with cdx dy we have then the integral of which is x = e-m** Q (y – amx) + e-*(y - anx); and from this u can be found. Eliminating u we find diz d’: -ac --- = 0: dx'dy dx? the integral of this equation is z = 0 (y) + xy (y) + €"""x(x); whence also u may be determined. (11) The equations for determining the small disturbances of an elastic medium in three dimensions are 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. Operate on each of these by respectively, and add : then 12 d 2 / d 2 du operate om each of these by (2) 6.). (6.) Us dy 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 d x HD ay + =O (2), where m2 = v2 + y®. Multiply the first equation by y and the second by r, and subtract; then - dog dog . |