I need help here because on case 2, my instructor said that I forgot to add the case 0<4c^2<1, but I am not sure how it would go with the solving order, could you help me please?
I need help here because on case 2, my instructor said that I forgot to add the case 0<4c^2<1, but I am not sure how it would go with the solving order, could you help me please?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Hello! I need help here because on case 2, my instructor said that I forgot to add the case 0<4c^2<1, but I am not sure how it would go with the solving order, could you help me please?
Transcribed Image Text:P
• Written• Assignment 7,
find the production solution for
u(x, y) = X w. Y (g)
S
X
X
so
+
and
XY+XY+X₁Y" = 0 Sevede bry X.Y
X'
X
+
Y = O
Y
so x (x) = C₁'l
u (x, y) ==
ta₁.e
=^
2
X" + 2Y = 0
so X" + X'
X
X
X₁ + X¹ = 2 ·X
and Y₁ = -2. Y so
X" + x² - 2²X=0 and Y" #2.Y=0
•cose 1:
270
let x = c²
X+X²-²x = 0
m-c²=0
m = -1 ± √(1-4X (-2²)
↓
2
-1 = √/1+4 c²; so - 1 - √/1+4c² and -1 + √/1+4=²²
2
2
2
Use separation of variables to
Jeu
+ du + d²u = 0
дуг
дж дуг
so My = X • Y| Mxx = X". Y| Myy = X.Y²
get
(-1 + √² +42²²) X
2
2
FO
"
so X +
X
11
and Y = -
Y
+ C₂0
m
m² + c
Yay) = A.cos (c.y) + B son (c.y)
p
X' =
X Y
+ C₂l
-^
ro m
(-1-√₁+422) X)
Y"+c².y=0
2
-y"
2
po
so we
ro
= 2
m
=± ci
contenue
Y
27 / ( A coo(2 g) + Branleg)) ✓
Transcribed Image Text:。
wratten edssegnment (part 2)
2 <0 so
cose 2:
so
from X" + X²-2.X=0
²+ m² + c ² = 0 so
m
m=
141c²
-11√1-4²
2
"
and from Y"+ 2. Y=0
and m²
and
=
2
then of L> 40²
so λ=-c²
1-40 40
so X (~) = e = ² ( ²₂ cos ( √² + 1/² | X + C₂ (
2
2
(-1+ √2-42²) X
2
.D.X
X(x) = C₁²e + C₂・é
x(x) = C₁+C₂ é
so x² + x' + c². X=0.
-1+√1-46²
2
m =
m = 1 c
them to 1-4c²>0
so
y"-².y=0
X(x) = C₁-e
and
ss
m = -1/₂ to
X(X) = (C₂₁₂ + C₂X). e
Cose 3:
2 = 0
X" + x² = 0 0 m² + m = 0
m/m + 11 = 0 0 mso and m= -1
+ C₂'l
X
so
1 = 4c² so 1-4c²20
so m²-e²=0
cy
Y(y) = A· + Bey
1-1-√√1-4-²
2
-1/2 and -1/₂ x
X
Y
som (x₁y) = ((C₂+₂x). e ^^) (A. + Beg)
y"+ X-Y=o
y" = o
y' = A
¡Y = AY+B
entigrat
agoen
X
Y
-X
→→ μ(x, g) = (C₁₂ + C₁₂ • e²^) (AY + B)
4
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