Basics of Fluid Mechanics, 2014a

586 APPENDIX A. MATHEMATICS FOR FLUID MECHANICS
Example A.7:
Solve the non-homogeneous equation
d 2 u
dt 2 − 5 du +6u = t + t2
dt
Solution
The homogeneous solution is
u(t) =c 1 e 2 t + c 1 e 3 t
the particular solution for t is
u(t) = 6 t +5
36
and the particular solution of the t 2 is
The total solution is
u(t) = 18 t2 +30t +19
108
u(t) =c 1 e 2 t + c 1 e 3 t + 9 t2 +24t +17
54
(1.VII.a)
(1.VII.b)
(1.VII.c)
(1.VII.d)
End Solution
A.2.5
Non–Linear Second Order Equations
Some of the techniques that were discussed in the previous section (first order ODE)
can be used for the second order ODE such as the variable separation.
A.2.5.1
Segregation of Derivatives
If the second order equation
can be written or presented in the form
f(u, ˙u, ü) =0
f 1 (u)˙u = f 2 (˙u)ü
(A.78)
then the equation (A.78) is referred to as a separable equation (some called it segregated
equations). The derivative of ˙u can be treated as a new function v and ˙v =ü. Hence,
equation (A.78) can be integrated
∫ u
f 1 (u)˙u =
u 0
∫ ˙u
f 2 (˙u)ü =
˙u 0
∫ v
f 2 (u) ˙v
v 0
(A.79)