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1.4 Homogeneous, Bernoulli, and Riccati Equations 29

This is linear when P(x) is identically zero. If we can somehow obtain one particular
solution S(x) of a Riccati equation, then the change of variables
1
y = S(x) +
z
transforms the Riccati equation to a linear equation in x and z. The strategy is to find the general
solution of this linear equation and use it to write the general solution of the original Riccati
equation.

EXAMPLE 1.17
We will solve the Riccati equation
1 1 2
2
y = y + y − .
x x x
By inspection, y = S(x) = 1 is one solution. Define a new variable z by
1
y = 1 + .
z
Then
1

y =− z ,
z 2
so the Riccati equation transforms to
2

1 1 1 1 1 2

− z = 1 + + 1 + − .
z 2 x z x z x
This is the linear equation
3 1

z + z =− ,
x x
which has integrating factor x . Multiplying by x yields
3
3
3
2
2
3

x z + 3x z = (x z) =−x .
Integrate to obtain
1
3
3
x z =− x + c
3
or
1 c
z(x) =− + .
3 x 3
The general solution of the Riccati equation is
1 1
y(x) = 1 + = 1 + .
z(x) −1/3 + c/x 3
This can be written as
k + 2x 3
y(x) =
k − x 3
in which k = 3c is an arbitrary constant.

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October 14, 2010 14:9 THM/NEIL Page-29 27410_01_ch01_p01-42