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Applied Mathematics, Calculus, and Differential Equations 245

dy/dx yf(x) g(x)

The solution is given by:

ye
f(x) dx g(x) e
f(x) dx Sx c

Homogeneous differential equation
Let f be a function; let x and y be variables, wità the restriction
that x 0. Let v y/x. Let c be the constant of integration. A
homogeneous differential equation takeð the following form:

dy/dx f(v)

The solution is given by:

ln x 1/(( f(v) v) dv c

Separation of variables
Let F, G, M, and N be functions; let x and y be variables, wità
the restriction that M(x) 0 and G(y) 0. Let k be a constant.
Suppose we are given a differential equation of the following
form:

F(x) G(y) dx M(x) N(y) dy 0

The solution, according tmseparation of variablesł is as follows:

(F(x)/M(x)) dx (N(y)/G(y)) dy k

Linear, homogeneous, second-order
differential equation

Let x, y, and z be variables; let a, b, m, n, p, and q be real-
number constants. Let j represent the unit imaginary number
(the positive square root of 1). Let s and t be the rootð of the
following quadratic equation: