The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 246 Wednesday, February 4, 2004 12:51 PM

246 The Mathematics of Financial Modeling and Investment Management

CLOSED-FORM SOLUTIONS OF ORDINARY

DIFFERENTIAL EQUATIONS

Let’s now consider the methods for solving two types of common differ-

ential equations: equations with separable variables and equations of lin-

ear type. Let’s start with equations with separable variables. Consider the

equation

-------= fx ()

dy

()gy

dx

This equation is said to have separable variables because it can be writ-

ten as an equality between two sides, each depending on only y or only

x. We can rewrite our equation in the following way:

dy

()dx

gy

-----------= fx

()

This equation can be regarded as an equality between two differentials

in y and x respectively. Their indefinite integrals can differ only by a

constant. Integrating the left side with respect to y and the right side

with respect to x, we obtain the general solution of the equation:

∫

----------dy- =

gy() ∫fx()d x+ C

For example, if g(y) ≡y, the previous equation becomes

------ = fx

dy

()dx

y

whose solution is

yd

∫------= ∫fx ()xd + C ⇒y = A exp( fx

y

()xd + C ⇒log y = ∫f x ∫()xd )

where A = exp(C).

A differential equation of this type describes the continuous com-

pounding of time-varying interest rates. Consider, for example, the

growth of capital C deposited in a bank account that earns the variable

but deterministic rate r = f(t). When interest rates Ri are constant for dis-