Page 48 - Advanced engineering mathematics

28 CHAPTER 1 First-Order Differential Equations

In about 1696, Leibniz showed that, if α = 1, the Bernoulli equation transforms to a linear
equation with the change of variable
v = y 1−α .
This is routine to verify in general. We will see how this works in an example.

EXAMPLE 1.16
We will solve the Bernoulli equation
1
2 3
y + y = 3x y .
x
2
Here P(x) = 1/x, R(x) = 3x , and α = 3. Let
−2
v = y 1−α = y .
Then y = v −1/2 ,so
1
v ,

y (x) =− v −3/2
2
and the differential equation becomes
1 1
2 −3/2
v + v
− v −3/2 −1/2 = 3x v .
2 x
3/2
Upon multiplying by −2v , we obtain the linear equation
2
2

v − v =−6x .
x
This has integrating factor
−2
−2
e −(2/x)dx = e ln(x ) = x .
−2
Multiply the differential equation by x :
−3
−2
x v − 2x v =−6.
This is
−2

(x v) =−6,
and an integration yields
−2
x v =−6x + c.
Then
2
3
v =−6x + cx .
In terms of y, the original Bernoulli equation has the general solution
1 1
y(x) = √ = √ .
2
v(x) cx − 6x 3
1.4.3 The Riccati Equation
A differential equation of the form
2

y = P(x)y + Q(x)y + R(x)
is called a Riccati equation.

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October 14, 2010 14:9 THM/NEIL Page-28 27410_01_ch01_p01-42