Basics of Fluid Mechanics, 2014a

A.2. ORDINARY DIFFERENTIAL EQUATIONS (ODE) 589
after integration
u du
dt = k 1
(A.89)
Further rearrangement and integration leads to the solution which is
u 2
2 k 1
= t + k 2 (A.90)
For non–homogeneous equation they can be integrated as well.
Example A.9:
Show that the solution of
u
( d 2 u
dt 2 )
+
( ) 2 du
+ u =0 (1.IX.a)
dt
is
−
√
3
∫
√
3
∫
u
√
3 k1 − u 3 du
√
2
= t + k 2
(1.IX.b)
u
√
3 k1 − u 3 du
√
2
= t + k 2
(1.IX.c)
A.2.6
Third Order Differential Equation
There are situations where fluid mechanics 7 leads to third order differential equation.
This kind of differential equation has been studied in the last 30 years to some degree.
The solution to constant coefficients is relatively simple and will be presented here.
Solution to more complicate linear equations with non constant coefficient (function of
t) can be solved sometimes by Laplace transform or reduction of the equation to second
order Olivier Vallee 8 .
The general form for constant coefficient is
d 3 u
dt 3 + a d2 u
dt 2 + b du
dt + cu=0
(A.91)
The solution is assumed to be of the form of e st which general third order polonium.
Thus, the general solution is depend on the solution of third order polonium. Third
7 The unsteady energy equation in accelerated coordinate leads to a third order differential equation.
8 “On the linear third-order differential equation” Springer Berlin Heidelberg, 1999. Solving Third
Order Linear Differential Equations in Terms of Second Order Equations Mark van Hoeij