Advanced High-School Mathematics

310 CHAPTER 5 Series and Differential Equations

2 x^2

dy

dx

= x^2 +y^2

can be reduced to the form (5.2) by dividing both sides by 2x^2 :

dy

dx

=

x^2 +y^2

2 x^2

=

1

2

ñ

1 +

Çy

x

åô

.

Settingv=yxas above reduces the above ODE to

x

dv

dx

+v =

1

2

(1 +v^2 );

that is,

2 x

dv

dx

= v^2 − 2 v+ 1.

After separating the variables, we arrive at the equation

∫ 2 dv

(v−1)^2

=

∫ dx

x

.

Integrating and simplifying yields

v = 1−

2

ln|x|+ 2C

.

Replacevbyy/x, setc= 2Cand arrive at the final general solution

y = x−

2 x

ln|x|+c

.

We define a function F(x,y) to be homogeneousof degree k if
for all real numberstsuch that (tx,ty) is in the domain ofF we have
F(tx,ty) = tkF(x,y). Therefore, the function F(x,y) = x^2 +y^2 is
homogeneous of degree 2, whereas the function F(x,y) =

√

x/y is
homogeneous of degree−^12.

A first-orderhomogeneousODE is of the form

M(x,y)

dy

dx

+N(x,y) = 0,