Bernoulli's Differential Equation

equation

x

′

y

(x)+y(x)-x

2

y(x)

c

0

x

′

y

(x)+y(x)-x

2

y(x)

z(x)-x

′

z

(x)-x

Bernoulli's differential equation has the form

y'+P(x)y=Q(x)

n

y

where

n≠0

or

1

.

Dividing by

n

y

, we get

-n

y

y'+P(x)

1-n

y

=Q(x)

.

Substituting

z=

1-n

y

and

z'=(1-n)

-n

y

y'

, we get the linear differential equation

z'/(1-n)+P(x)z=Q(x)

.

This Demonstration shows Bernoulli's equation and solutions for a few choices of

P(x)

and

Q(x)

. Each equation has a family of solutions parametrized by the parameter

c

.