Basics of Fluid Mechanics, 2014a

582 APPENDIX A. MATHEMATICS FOR FLUID MECHANICS
Solution
The transformation is
v = u 2
Using the definition (1.III.b) equation (1.III.a) becomes
dv
dt
1−p
{}}{
−2 t 2 v =
The homogeneous solution of equation (1.III.c) is
1−p
{}}{
−2 sin(t)
(1.III.b)
(1.III.c)
u(t) =ce −t3
3 (1.III.d)
And the general solution is
u = e −t3 3
⎛
⎞
private solution
{ }} {
⎜∫
⎝
e t3 ⎟
⎠
3 sin (t) dt +c
(1.III.e)
End Solution
A.2.3.1
Homogeneous Equations
Homogeneous function is given as
du
= f(u, t) =f(au,at)
dt (A.63)
for any real positive a. For this case, the transformation of u = vttransforms equation
(A.63) into
t dv
dt + v = f(1,v)
(A.64)
In another words if the substitution u = vtis inserted the function f become a function
of only v it is homogeneous function. Example of such case u′ = ( u 3 − t 3) /t 3 becomes
u′ = ( v 3 +1 ) . The solution is then
∫
dv
ln |t| =
f(1,v) − v + c
(A.65)
Example A.4:
Solve the equation
du
( u
) ( u 4
dt = sin − t 4 )
+
t t 4
(1.IV.a)