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algebraic angle applies arbitrary constants arbitrary function arise auxiliary equation axis Chapter classes of equations complementary function complete solution condition constant coefficients constant of integration corresponding dependent variable derivatives determine dx dx dx dz dy dx dy dy eliminate envelope equa equation becomes equation with constant exact exact differential existence theorem family of curves Find the differential Find the orthogonal finite given gotten Hence higher order homogeneous homogeneous function independent integral curves integrating factor left-hand member Let the student linear differential equation linear equation method obviously ordinary differential equations orthogonal trajectories partial differential equation particular integral particular solution plane primitive quadratures readily result Riccati equation right-hand member second order Section singular solution Substituting suppose tangent tion total differential equation transformation variables are separated velocity whence zero
Page 167 - S s2 dm are the moments of inertia of the body with respect to the planes of yz, zx and xy, respectively.
Page 5 - ... of various orders. The order of a differential equation is the order of the highest derivative which occurs in it. A solution of a differential equation is any relation between the variables, which, when substituted in the given equation, will satisfy it. The general solution of an ordinary differential equation of the nth order will contain n arbitrary constants.
Page 171 - Thus, we see that the general solution of a differential equation of the nth order must contain n and only n independent arbitrary constants.
Page 124 - ... condenser. With a steady current /' maintained in the field coils of the dynamometer, arranged as in Fig. 2, a condenser is discharged through the moving coil of the instrument. As before, the instantaneous torque acting on the moving coil of the dynamometer during discharge is given by (22) where q is the quantity of electricity in the condenser at any instant. Multiplying by dt and replacing Tdt by its value from equation (12), and integrating, we get £=»tnai /•«=<> du=-gl> (23) __ 0 */q=Q...
Page 38 - II. Orthogonal Trajectory A curve that intersects every member of a family of curves according to some law is called a trajectory of the family. If two families of curves such that every member of the either family cuts each member of the other family at right angles, then they are called Orthogonal trajectories (OT) of each other. In stead of right angles, if they cut each other at constant angle then they are called Isogonal trajectories.
Page 93 - A linear differential equation is one which is of the first degree in the dependent variable and all of its derivatives. Its general type is fi\ W ах" ах ' dxn where X0, X”, X2, •••, Х„, X are functions of x or constants.
Page 205 - MONGE'S METHOD (NON LINEAR EQUATION OF THE SECOND ORDER) Let the equation be Rr + Ss + Tt = V - (1) where R, S, T, V are functions of x, y, z, p and q.