Lecture Notes in Differential Equations - Bruce E. Shapiro

412 APPENDIX C. SUMMARY OF METHODS
Integrating Factors
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 ∫ 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 ∫ 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:
dz
F (z) − z = dt
t
Bernoulli Equations
A Bernoulli equation has the form
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