Basics of Fluid Mechanics, 2014a

A.2. ORDINARY DIFFERENTIAL EQUATIONS (ODE) 581
Example A.2:
Find the solution for a typical problem in fluid mechanics (the problem of Stoke flow or
the parachute problem) of
dy
dx + y =1
Solution
Substituting m(x) =1and g(x) =1into equation (A.59) provides
y = e −x (e x + c) =1+ce −x
End Solution
A.2.3
Non–Linear Equations
Non-Linear equations are equations that the power of the function or the function
derivative is not equal to one or their combination. Many non linear equations can be
transformed into linear equations and then solved with the linear equation techniques.
One such equation family is referred in the literature as the Bernoulli Equations 5 . This
equation is
non–linear
part
du
{}}{
dt + m(t)u = n(t) u p
(A.60)
The transformation v = u 1−p turns equation (A.60) into a linear equation which is
dv
+(1− p) m(t) v =(1− p) n(t)
dt (A.61)
The linearized equation can be solved using the linear methods. The actual solution is
obtained by reversed equation which transferred solution to
u = v (p−1)
(A.62)
Example A.3:
Solve the following Bernoulli equation
du
dt + t2 u = sin(t) u 3
(1.III.a)
5 Not to be confused with the Bernoulli equation without the s that referred to the energy equation.