An Elementary Treatise on Differential Equations |
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Common terms and phrases
arbitrary constants arbitrary function auxiliary equation axis c-discriminant c₁ c₂ complementary function complete solution constant coefficients constant of integration coördinates corresponding d²y d³y degree dependent variable derivatives dx dx dx dy dx Ex dx² dy dx dy Ex eliminating envelope equa equation with constant existence theorem finite given gotten Hence independent integral curves integrating factor involving left-hand member Let the student linear differential equation linear equation locus method ordinary differential equations orthogonal trajectories partial differential equation particular integral plane quadratures result Riccati equation right-hand member satisfy second order Section singular solution solved Substituting tangent tion transformation velocity whence y₁ zero әм ди ди др ду ду ду дх дх ду მა მე მი
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 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 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 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.