Page 57 - Standard Handbook Of Petroleum & Natural Gas Engineering

46 Mathematics

equation involves an unknown function of two or more independent variables,
and its partial derivatives. The general solution of a differential equation of
order n is the set of all functions that possess at least n derivatives and satisfy
the equation, as well as any auxiliary conditions.

Methods of Solving Ordinary Differential Equations

For first-order equations, if possible, separate the variables, integrate both sides,
and add the constant of integration, C. If the equation is homogeneous in x and y,
the value of dy/dx in terms of x and y is of the form dy/dx = f(y/x) and the
variables may be separated by introducing new independent variable v = y/x
and then

dv
x-+v = f(v)
dx
The expression f(x,y)dx + F(x,y)dy is an exact difj-erential if

Then, the solution of f(x,y)dx + F(x,y)dy = 0 is
jf(x,y)dx + [F(x,y) - IP dxldy = C

or
hx,y)dy + [f(x,y) - h' dyldx = C

A linear differential equation of the first order such as
dy/dx + f(x) y = F(x)

has the solution
y = e-'[ jePF(x)dx + C] where P = If(x)dx

In the class of nonlinear equations known as Bernoulli's equations, where
dy/dx + f(x) y = F(x) y"
substituting y'-" = v gives

dv/dx + (1 - n)f(x) v = (1 - n)F(x) [n # 0 or 11
which is linear in v and x. In Clairaut's equations

y = xp + f(p) where p = dy/dx,
the solution consists of the set of lines given by y = Cx + f(C), where C is any
constant, and the curve obtained by eliminating p between the original equation
and x + f'(p) = 0 [l].