Advanced High-School Mathematics

308 CHAPTER 5 Series and Differential Equations

C(x) =xn+an− 1 xn−^1 +···+a 1 x+a 0

has a real zeroα, i.e., thatC(α) = 0. Show that a solution of the

ODE (5.1) isy=eαx.

5.5.2 Separable and homogeneous first-order ODE

Most students having had a first exposure to differential and integral
calculus will have studiedseparablefirst-order differential equations.
These are of the form

dy

dx

= f(x)g(y)

whose solution is derived by an integration:

∫ dy

g(y)

=

∫

f(x)dx.

Example 1. Solve the differential equation

dy

dx

=− 2 yx.

Solution. From

∫ dy

y

= −

∫

2 xdx.

we obtain

ln|y| = −x^2 +C,

where C is an arbitrary constant. Taking the natural exponential of
both sides results in |y| = e−x

(^2) +C
= eCe−x
2

. However, if we define
K=eC, and if we allowK to take on negative values, then there is no
longer any need to write|y|; we have, therefore the general solution

y = Ke−x

2

,