By Norman W. Loney
DIFFERENTIAL EQUATIONS creation traditional Differential Equations version improvement References FIRST-ORDER usual DIFFERENTIAL EQUATIONS Linear Equations additional info on Linear Equations Nonlinear Equations challenge Setup difficulties References LINEAR SECOND-ORDER AND platforms OF FIRST-ORDER traditional DIFFERENTIAL EQUATIONS creation basic ideas of the Homogeneous Equation Homogeneous Equations with consistent Coefficients Nonhomogeneous Equations Variable Coefficient difficulties replacement equipment precis functions of Second-Order Differential Equations. Read more...
summary: DIFFERENTIAL EQUATIONS advent usual Differential Equations version improvement References FIRST-ORDER usual DIFFERENTIAL EQUATIONS Linear Equations more information on Linear Equations Nonlinear Equations challenge Setup difficulties References LINEAR SECOND-ORDER AND platforms OF FIRST-ORDER traditional DIFFERENTIAL EQUATIONS creation basic options of the Homogeneous Equation Homogeneous Equations with consistent Coefficients Nonhomogeneous Equations Variable Coefficient difficulties substitute tools precis functions of Second-Order Differential Equations
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Additional info for Applied Mathematical Methods for Chemical Engineers, Second Edition
Always check for the involvement of some physical law or principle (mass, momentum, or energy balances). Include a consistent set of units (same units in each term). Appropriately include given conditions (initial or boundary). Check the mode of operation (steady or unsteady process). Below are some examples for which the straightforward application of mass or energy balance is sufficient to set up the differential equation. 6 Transient Behavior of an Air-Cooling System  Consider an engine that generates heat at a rate of 8530 Btu/min.
Thus, m(x) must satisfy m(x)p(x)y ¼ m0 (x)y 22 Applied Mathematical Methods for Chemical Engineers or m0 (x) ¼ p(x), m(x) > 0 m(x) then ln m(x) ¼ ðx p(t) dt or m(x) ¼ exp ðx ! 29 by m(x). 32 with respect to x and solving for y gives 1 y¼ m(x) ðx ! m(s)g(s) ds þ c (2:33) Further, Since p is continuous for a < x < b, it follows that m is defined in this interval and is a nonzero differentiable function. 32 is justified. 32. 29. 30, determines the integration constant c uniquely. Sometimes nonlinear equations can be reduced to linear ones by a substitution.
7 are continuous on a < x < b, then for any twice-differentiable function w on a < x < b, the linear differential operator L is defined to mean L[f] ¼ f00 þ pf0 þ qf (3:8) L D2 þ pD þ q (3:9) for where D is the derivative operator. 7 becomes L[y] ¼ y00 þ p(x)y0 þ q(x)y ¼ 0 (3:10) The use of the operator L here reduces the task of integrating the linear second-order differential equation as will be seen below. However, before the integration process can be employed, a few more definitions are needed.
Applied Mathematical Methods for Chemical Engineers, Second Edition by Norman W. Loney