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algebraic expression arbitrary constants arbitrary function auxiliary equation CHAPTER constant and equal constant coefficients continuous function cosec Definition denotes Determine the curve differentiation with respect dv dv dv dv dy dx dx dy dx dy dz dy dx dy dz dz dz dx dx Eliminate equa equation becomes equation Mdx equations dx dy Example following exercises hand member zero independent variable indicial equation infinite series Laplace's Equation left hand member Legendre's Equation linear differential equation Mdx f mn)y Multiply necessary and sufficient ordinary differential equation ordinary linear differential original equation partial differential equation partial fractions particular solutions positive integer power series preceding article regular point right hand member roots satisfies the equation second order single valued solu Solve the equation Substitute subtangent sufficient condition Suppose symbolic factors theorem tion valued and continuous viewed as algebraic
Page 79 - Rdz = 0, where P, Q and R are functions of x, y and z, and do not contain the arbitrary constant c.
Page 59 - In each of the six following exercises, find the equation of the elastic curve of the beam from the given differential equation, determining the constants of integration. Find also the deflection of the beam. In these equations, E is the modulus of elasticity, J is the moment of inertia of a cross section of the beam about a gravity axis in the section perpendicular to the applied forces, and I is the length of the beam. 15. The beam rests on supports at its ends. It is supposed weightless with a...
Page 22 - Mdx + Ndy = 0 where M and N are functions of x and y and do not contain derivatives.
Page 85 - This is a linear differential equation of the second order with constant coefficients and right hand member zero.
Page 53 - The result when the operator is applied to the sum of a number of functions is equal to the sum of the results found when the operator is applied to each of the functions separately.
Page 79 - Bz' respectively. Conversely, however, an equation of the form Pdx + Qdy + Rdz = 0 where P, Q and R are functions of x, y and z, does not necessarily give rise to a solution of the form <f>(x, y, 2) = c.
Page 7 - ORDER AND DEGREE OF A DIFFERENTIAL EQUATION The order of a differential equation is the order of the highest differential co-efficient present in the equation. Consider f) i -f> / j fi*. f* ft** f* j £2
Page 41 - A typical linear equation of the first order is where P and Q are functions of x only or constants.
Page 26 - Ndy = 0 where M and N are functions of x and y, is said to be exact when there is a function u(x, y) such that du = Mdx -f- Ndy.
Page 36 - Determine the curve in which the angle between the radius vector and the tangent line is n-times the vectorial angle. Plot the curve when n = •£. 45. Determine the curve in which the polar subnormal is proportional to the sine of the vectorial angle. 46. Determine the curve in which the polar subtangent is proportional to the length of the radius vector. The equation for a circuit containing induction and resistance is...