## An Elementary Treatise on Differential Equations |

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### Common terms and phrases

algebraic angle applies arbitrary constants arbitrary function arise auxiliary equation axis Chapter complementary function complete solution condition constant coefficients constant of integration corresponding degree dependent variable derivatives dx dy dy dx dy dy dy dz eliminating envelope equa equation with constant exact exact differential existence theorem Find the differential Find the orthogonal find the particular finite given gotten Hence homogeneous homogeneous function integral curves integrating factor involving left-hand member Let the student linear differential equation linear equation method obviously operators ordinary differential equations orthogonal trajectories ox ay partial differential equation particular integral particular solution plane problem quadratures r-fold root readily result Riccati equation right-hand member second order Section singular solution solved Substituting tangent tion total differential equation transformation variables are separated velocity whence zero

### Popular passages

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.