Page 558 - Modelling in Transport Phenomena A Conceptual Approach

538 APPENDIX B. SOLUTIONS OF DIFFERENTIAL EQUATIONS

B.2.1 Special Case of a Second-Order Equation

A second-order ordinary differential equation of the form

(B.2-8)

where A is a constant, is frequently encountered in heat and mass transfer problems.
Since the roots of the characteristic equation are

m1,2 = f A (B.2-9)

the solution becomes
y = C1 exx + C2 e-'" (B.2-10)
Using the identities
,Ax + e-Xx ,Ax - e-Xx
coshAx= and sinhAx= (B.2-11)
2 2
Equation (B.2-10) can be rewritten as

y = C,* sinh Ax + Cz cosh Ax (B.2-12)

B.2.2 Solution of a Non-Homogenous Differential
Equation

Consider the second-order differential equation

(B.2-13)

If one solution of the homogeneous solution is known, i.e., say y = yl(x), then the
complete solution is (Murray, 1924)

Example B.5 Obtain the complete sobtion of the following nowhomogeneous
differential equation if one of the solutions of the homogeneous part is y1 = e2x.