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algebraic expression arbitrary constants arbitrary function auxiliary equation CHAPTER constant and equal constant coefficients continuous function cosec ct Va2 Definition denotes Determine the curve differentiation with respect dv dv dv dv dy dx dx dy dx dy dz dy dx dy dy dy dz dz dz dx dx Eliminate equa equation becomes equation Mdx equations dx dy Example following equations following exercises hand member zero independent variable indicial equation infinite series Laplace's Equation left hand member Legendre's Equation linear differential equation 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 Suppose symbolic factors theorem tion valued and continuous
Page 59 - E is the modulus of elasticity, / 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.
Page 79 - Sx' dy and 8z' 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 ф(ж, у, г) = с.
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 65 - ... 0, and D = d/dx. It is readily recognized from the fact that the powers of x and of D in any term of the operator are the same. Euler's equation may always be transformed into a linear equation with constant coefficients by changing the independent variable from x to z by the substitution x = tf, or z = Inz.
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 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 36 - Determine the curve in which the polar subnormal is proportional to the sine of the vectorial angle.