Page 47 - Advanced engineering mathematics

1.4 Homogeneous, Bernoulli, and Riccati Equations 27

or
du
x = f (u) − u.
dx
The variables u and x separate as
1 1
du = dx.
f (u) − u x
We attempt to solve this separable equation and then substitute u = y/x to obtain the solution of
the original homogeneous equation.

EXAMPLE 1.15

We will solve
y 2

xy = + y.
x
Write this as

2 y
y

y = + .
x x
With y = ux, this becomes
2

xu + u = u + u
or
2
xu = u .

The variables separate as
1 1
du = dx.
u 2 x
Integrate to obtain
1
− = ln|x|+ c.
u
Then
−1
u = .
ln|x|+ c
Then
−x
y = ,
ln|x|+ c
and this is the general solution of the original homogeneous equation.

1.4.2 The Bernoulli Equation

A Bernoulli equation is one of the form

y + P(x)y = R(x)y α
in which α is constant. This equation is linear if α = 0 and separable if α = 1.

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October 14, 2010 14:9 THM/NEIL Page-27 27410_01_ch01_p01-42