Lecture Notes in Differential Equations - Bruce E. Shapiro

82 LESSON 10. INTEGRATING FACTORS
Rearranging,
1 d
µ(y) dy µ(y) = N t(t, y) − M y (t, y)
= P (2) (y) (10.44)
M(t, y)
Multiply by dy and integrating,
∫
∫
1 d
µ(y) dy µ(y)dy = P (2) (y)dy (10.45)
Integrating and exponentiating
(∫
µ(y) = exp
)
P (2) (y)dy
(10.46)
as required by equation (10.29). (Case 2).
Case 3. If µ(t, y) = µ(z) is only a function of z = ty then
∂z
∂t = ∂(ty) = y
∂t
(10.47)
∂z
∂y = ∂(ty) = t
∂y
(10.48)
Since µ(z) is only a function of a single variable z we will denote
µ ′ (z) = dµ
dz
By the chain rule and equations (10.47),(10.48), and (10.49),
Using these results in (10.38),
(10.49)
∂µ
∂t = dµ ∂z
dz ∂t = µ′ (z)y (10.50)
∂µ
∂y = du ∂z
dz ∂y = µ′ (z)t (10.51)
µ(z)M y (t, y) + µ ′ (z)tM(t, y) = µ(z)N t (t, y) + µ ′ (z)yN(t, y) (10.52)
µ ′ (z) × (tM(t, y) − yN(t, y)) = µ(z) × (N t (t, y) − M y (t, y)) (10.53)
µ ′ (z)
µ(z) = N t(t, y) − M y (t, y)
tM(t, y) − yN(t, y)) = P (3) (z) (10.54)
Integrating and exponentiating,
(∫
µ(z) = exp
)
P (3) (z)dz
(10.55)