An Elementary Treatise on Differential Equations

D.C. Heath & Company, 1906 - Differential equations - 271 pages

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Page 38 - Find the curve in which the perpendicular from the origin upon the tangent is equal to the abscissa of the point of contact.
Page 229 - Determine the equation to the surface in which the coordinates of the point where the normal meets the plane of xy, are to each other as the corresponding coordinates. The equations to the normal are /y' dz df but &.£, TJO y wliere / represents an arbitrary function.
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 163 - S s2 dm are the moments of inertia of the body with respect to the planes of yz, zx and xy, respectively.
Page 167 - 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 201 - Rr + Ss + Tt = V, where R, S, T, V are functions of x, y, z, p, q, will be treated by Monge's method.
Page 120 - ... 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 89 - A linear differential equation is one which is of the first degree in the dependent variable and all of its derivatives.
Page 90 - Functions ul, u2, u3, . . . are said to be linearly independent if it is impossible to find constants Cl, C2, C3, . . . not all zero such that (7^+ 02M2 + CgMj + ... =0 identically.
Page 230 - 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.