Differential Equations of First Order and First Degree 2-21
Using (1,)
λ
as multipliers, each ratio
()
()()
dyx
alxbmycn
λ
λλλ
+
=
+++++
(2.44)
Now, we determine
λ
such that
al
blm
λ
λ
+
=
+
or
2
()0mbla
λλ
+−−= (2.45)
which is a quadratic in l and has two real distinct roots l
1
, l
2
if
2
()40.blamΔ=−+> More specifi cally
2
12
,()()42(0).blblammm
λλ
⎡⎤
=−−±−+≠
⎣⎦
Also,
(1,2)
i
i
i
al
i
bm
λ
λ
λ
+
==
+
∴ each ratio
12
111222
()()
11
dyxdyx
bmyxkbmyxk
λλ
λλλλ
++
==⋅
++++++
(2.46)
;1,2
i
i
i
cn
ki
bm
λ
λ
⎛⎞
+
==
⎜⎟
+
⎝⎠
Integrating and dropping logs, the general solution of equation
(2.34) in this case may be written as
21
1122
()()
bmbm
yxkCyxk
λλ
λλ
++
++=++ (2.47)
Note 2.1.21 ...
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