Modeling Tools for Environmental Engineers and Scientists

c. Second-order equation with equidimensional coefficients: when the ODEcan be expressed in the form:2 y d dx2 a x d y dxxb2 y = 0a trial solution of the form y = cx s , where s is found from:(a – 1) ± (a – 1)s = – 2 – 4b(3.19)2giving the solution asy = b 1 x s1 + b 2 x s2 if s1 ≠ s2 (3.20a)y = b 1 x s + b 2 x s ln x if s1 = s2 = s (3.20b)d. Second-order equation with constant coefficients: when the ODE can beexpressed in the form:2 y d dx2 a d y by = 0dxa trial solution of the form y = be sx , where s is found from:giving the solution asWorked Example 3.3a ± as = – 2 – 4b (3.21)2y = b 1 e s1x + b 2 e s2x if s1 ≠ s2 (3.22a)y = b 1 e sx + b 2 xe sx if s1 = s2 = s (3.22b)The buildup of the concentration, C, in a lake due to a new step waste loadinput can be described by the following equation (to be derived in Chapter 6): d C 1.23C = 2.0dtThe initial concentration of the waste material in the lake = 0. Develop a solutionto the above ODE to describe the concentration in the lake as a functionof time.© 2002 by CRC Press LLC