23.2 Bernouli Equations
Some kinds of nonlinear equations can be changed to get a linear equation. An equation of the
is called a Bernouli equation .
The trick is to define a new variable, z = y1−α. Then yαz = y and so z′ =
Now this is a linear equation for z. Solve it and then use the transformation to find y.
Example 23.2.1 Solve y′ + y = ty3.
You let z = y−2 and make the above substitution. Thus zy3 = y and
and so −
Hence, cancelling the y3,z′−
When you get this far, it is a good idea to check and see if it works. After all, this is the point of the
manipulations, to get the answer. If you get the answer, then if there is a mistake, it is no longer terribly
so it appears to work.
The following procedure gives a summary of the above.
Procedure 23.2.2 To solve the Bernouli equation
do the following:
- Change the variable. Let z = y1−α. Then z′ =
y−αy′,yαz = y.
- Place in the equation.
- Cancel the yα and solve the linear equation for z.