29.2. BERNOULI EQUATIONS 545

29.2 Bernouli EquationsSome kinds of nonlinear equations can be changed to get a linear equation. An equation ofthe form

y′+a(t)y = b(t)yα

is called a Bernouli equation1. The trick is to define a new variable, z = y1−α . Then yα z = yand so z′ = (1−α)y−α y′ which implies 1

(1−α)yα z′ = y′. Then

1(1−α)

yα z′+a(t)yα z = b(t)yα

and soz′+(1−α)a(t)z = (1−α)b(t) .

Now this is a linear equation for z. Solve it and then use the transformation to find y.

Example 29.2.1 Solve y′+ y = ty3.

You let z = y−2 and make the above substitution. Thus zy3 = y and

z′ = (−2)y−3y′, y′ =−12

y3z′

and so − 12 y3z′+ y3z = ty3. Hence, cancelling the y3,z′−2z = (−2) t. Then

ddt

(e−2tz

)=−2te−2t

and soe−2tz = te−2t +

12

e−2t +C

and soy−2 = z = t +

12+Ce2t

and soy2 =

1t + 1

2 +Ce2t.

When you get this far, it is a good idea to check and see if it works. After all, this isthe point of the manipulations, to get the answer. If you get the answer, then if there is amistake, it is no longer terribly relevant.

2yy′ =ddt

(1

t + 12 +Ce2t

)=− 8Ce2t +4

(2t +2Ce2t +1)2

y′ =− 8Ce2t +4

2y(2t +2Ce2t +1)2

1This is named after Jacob Bernoulli (1654-1705), one of a whole family of Swiss mathematicians. Otherswere Johann I and II Daniel, and Nicolaus.