The Computable Differential Equation Lecture ... - Bruce E. Shapiro

194
Integrating Factors
CHAPTER 10.
APPENDIX ON ANALYTIC METHODS
(MATH 280 IN A NUTSHELL)
An integrating factor µ for the differential equation
satisfies
If
M(t, y)dt + N(t, y)dy = 0
∂(µ(t, y)M(t, y))
∂y
=
P (t, y) = M y − N t
N
∂(µ(t, y)N(t, y))
∂t
is only a function of t (and not of y) then µ(t) = e R P (t)dt is an integrating factor.If
Q(t, y) = N t − M y
M
is only a function of y (and not of t) then µ(t) = e R Q(t)dt is an integrating factor.
Homogeneous Equations
An equation is homogeneous if has the form
y ′ = f(y/t)
To solve a homogeneous equation, make the substitution y = tz and rearrange the
equation; the result is separable:
Bernoulli Equations
A Bernoulli equation has the form
dz
F (z) − z = dt
t
y ′ (t) + p(t)y = q(t)y n
for some number n. To solve a Bernoulli equation, make the substitution
u = y 1−n
The resulting equation is linear and
y(t) =
[ ( ∫ 1
C +
µ
)] 1/(1−n)
µ(t)(1 − n)q(t)dt
where
( ∫
µ(t) = exp (1 − n)
)
p(t)dt
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007