Lecture Notes in Differential Equations - Bruce E. Shapiro

83
as required by equation (10.31). (Case 3)
Case 4. If µ(t, y) = µ(z) is only a function of z = y/t then
By thee chain rule
where µ ′ (z) = du/dz. Equation (10.38) becomes
µ(z)M y (t, y) + µ′ (y)
t
Rearranging and solving for µ ′ (z)
∂z
∂t = ∂ y
∂t t = − y t 2 (10.56)
∂z
∂y = ∂ y
∂y t = 1 t
(10.57)
∂µ
∂t = dµ ∂z
dz ∂t = y −µ′ t 2 (10.58)
∂µ
∂y = dµ ∂z
dz ∂y = µ′
t
(10.59)
× M(t, y) = µ(y)N t (t, y) +
(−µ ′ (z) y )
t 2 × N(t, y)
( yN(t, y)
µ(z)(M y (t, y) − N t (t, y)) = −µ ′ (z)
t 2 +
)
M(t, y)
t
(10.60)
(10.61)
= − µ′ (z)
t 2 (yN(t, y) + tM(t, y)) (10.62)
µ ′ (z)
µ(z) = t2 (N t (t, y) − M y (t, y)
= P (4) (z)
yN(t, y) + tM(t, y)
(10.63)
Integrating and exponentiating,
(∫
µ(z) = exp
)
P (4) zdz
(10.64)
as required by equation (10.33). (Case 4)
Case 5. If µ(t, y) = µ(z) is only a function of z = t/y then
∂z
∂t = ∂ t
∂t y = 1 y
(10.65)
∂z
∂y = ∂ t
∂y y = − t
y 2 (10.66)