Consider the system: X1x3 + x₂x² = 0 |x₁x³ + x²x = 0 1. Can we solve for x3 and x4 as functions of ₁ and ₂ near (1,-1, 1, −1)? Can you use the Implicit Function Theorem to justify your answer? Explain why. 2. Can we solve for 3 and 4 as functions of x₁ and x2 near (0, 0, 0, 0)? Can you use the Implicit Function Theorem to justify your answer? Explain why. 3. Solve for 3 and 4 in terms of x₁ and x2. These formulas may help to understand whether or not we can solve for x3 and 4 as functions of ₁ and 2 near (C₁, C2, C3, C4).
Consider the system: X1x3 + x₂x² = 0 |x₁x³ + x²x = 0 1. Can we solve for x3 and x4 as functions of ₁ and ₂ near (1,-1, 1, −1)? Can you use the Implicit Function Theorem to justify your answer? Explain why. 2. Can we solve for 3 and 4 as functions of x₁ and x2 near (0, 0, 0, 0)? Can you use the Implicit Function Theorem to justify your answer? Explain why. 3. Solve for 3 and 4 in terms of x₁ and x2. These formulas may help to understand whether or not we can solve for x3 and 4 as functions of ₁ and 2 near (C₁, C2, C3, C4).
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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Question
Transcribed Image Text:Consider the system:
X1X3 + x2x² = 0
|x₁x³ + x² = 0
X1 and x2 near (1, 1, 1,-1)?
1. Can we solve for x3 and 4 as functions of
Can you use the Implicit Function Theorem to justify your answer?
Explain why.
2. Can we solve for 3 and 4 as functions of x₁ and x2 near (0, 0, 0, 0)?
Can you use the Implicit Function Theorem to justify your answer?
Explain why.
3. Solve for x3 and 4 in terms of ₁ and x2. These formulas may help to
understand whether or not we can solve for 3 and 4 as functions of
₁ and 2 near (C₁, C2, C3, C4).
![ex Let F: IR→IR² where
F(X₁1X₁₁ X3 X 4) = (x
Can we solve [F(3)= 5 = (0,0) ER²
for
and
X3 and
X₂
Xy as tons of X₁
4
2
Imp. Fen. Thm. Let F:IR" - R"
F₂
X₁ X ₂ + x3 x4)
F = (F₁,F₂), F(c) =Ō = (0,0) for some
C = (²₁1²2, C3, C4) ER", and
Ə(F₁,F₂)
2(x3, x4)
Then 3
be c',
- (c) = det
93: N-IR and Øy: N→ IR S.t.
x3 = 3 (x₁₁x₂) + xy = $4(x₁1x₂)
solve F(X₁, X₂, X3, Xy) = (X₁, X₂) EN
Øz (13 (2)=C3 and $4((1162) = (4.
and
Let
So
3
#0.
c = (',!,!,-¹).
2 (F₁, F₂).
2 [x3, x4)
~X₂ X3 = -10.
=
JF₂
2x1
det
JF, 26
Эхз даху
✓
since
Also F is CDF is the
DF=
- X₂
Ху
CaF₂
[D3 F₁(x) D₂ F₁(x)]
[Dz F₂ (x) ; D 4 F₂(x)]
So by IFT, we can solve for xz + xy
a nhood N of (C₁, 5₂) and C' fons as fons. of x₁ * x₂ on some nhood N
F(X₁, X₁, X₂, Xu) = (0,0)
get X z = 2 / ² = Øy (X₁Xx₂).
of (²₁₂₂): (1.1) s.t.
Set F₁ = x₁-x₂x3 =
3
Set F₂=X,X₂+X3 Xy=
Ø₂(x+, *2) $4 (X, 5X2) get X4=;
=
F(x, X₂, 3, 4) = 0
[1])
і Xyl X3
Also F(c)= F(1, 1, 1, -1) = (0, 0)
2 F₂
аху
2x3
the 2xy matrix](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79599c56-a340-49a0-b0ff-829b3947a798%2F4b450426-7e07-465f-b378-ffb4ef29afd2%2F3jbxwtc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:ex Let F: IR→IR² where
F(X₁1X₁₁ X3 X 4) = (x
Can we solve [F(3)= 5 = (0,0) ER²
for
and
X3 and
X₂
Xy as tons of X₁
4
2
Imp. Fen. Thm. Let F:IR" - R"
F₂
X₁ X ₂ + x3 x4)
F = (F₁,F₂), F(c) =Ō = (0,0) for some
C = (²₁1²2, C3, C4) ER", and
Ə(F₁,F₂)
2(x3, x4)
Then 3
be c',
- (c) = det
93: N-IR and Øy: N→ IR S.t.
x3 = 3 (x₁₁x₂) + xy = $4(x₁1x₂)
solve F(X₁, X₂, X3, Xy) = (X₁, X₂) EN
Øz (13 (2)=C3 and $4((1162) = (4.
and
Let
So
3
#0.
c = (',!,!,-¹).
2 (F₁, F₂).
2 [x3, x4)
~X₂ X3 = -10.
=
JF₂
2x1
det
JF, 26
Эхз даху
✓
since
Also F is CDF is the
DF=
- X₂
Ху
CaF₂
[D3 F₁(x) D₂ F₁(x)]
[Dz F₂ (x) ; D 4 F₂(x)]
So by IFT, we can solve for xz + xy
a nhood N of (C₁, 5₂) and C' fons as fons. of x₁ * x₂ on some nhood N
F(X₁, X₁, X₂, Xu) = (0,0)
get X z = 2 / ² = Øy (X₁Xx₂).
of (²₁₂₂): (1.1) s.t.
Set F₁ = x₁-x₂x3 =
3
Set F₂=X,X₂+X3 Xy=
Ø₂(x+, *2) $4 (X, 5X2) get X4=;
=
F(x, X₂, 3, 4) = 0
[1])
і Xyl X3
Also F(c)= F(1, 1, 1, -1) = (0, 0)
2 F₂
аху
2x3
the 2xy matrix
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