Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 5
Bernoulli Equations
The Bernoulli differential equation has the form
y ′ + p(t)y = y n q(t) (5.1)
which is not-quite-linear, because of the factor of y n on the right-hand side
of equation 5.1 would be linear. It does reduce to a linear equation, when
either n = 0 or n = 1. In the first case (n = 0), we have
y ′ + p(t)y = q(t) (5.2)
which is a general first-order equation.
Bernoulli equation becomes
In the second case (n = 1)the
which can be rearranged to give
y ′ + p(t)y = q(t)y (5.3)
y ′ = (q(t) − p(t))y (5.4)
This can be solved by multiplying both sides of equation 5.4 by dt/y, and
integrating:
∫ ( ) ∫ dy
dt = (q(t) − p(t))dt (5.5)
dt
∫
y = (q(t) − p(t))dt + C (5.6)
For any other value of n, Bernoulli equations can be made linear by making
the substitution
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