GivenS.t.JLJ QI1L (Q₁, Q₂ ) = U - λ (Y - P₁ Q₁ - P2 Q2)2JQ 2ах1-4U (Q₁, Q₂) = Q₁, * Q ₂²иса,,P2Q2Y = P₁ Q₁ +алSolve==.Q₁ * =Q₂² =

Given such that Consider the Lagrange function We need to compute the derivatives and find the…

Answered: Given U (Q₁, Q₂ ) = Q ₁ * Q ₂² 1-d Y =…

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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Related questions

Question

Given
S.t.
JL
J QI
1
L (Q₁, Q₂ ) = U - λ (Y - P₁ Q₁ - P2 Q2)
2
JQ 2
ах
1-4
U (Q₁, Q₂) = Q₁, * Q ₂²
иса,,
P2
Q2
Y = P₁ Q₁ +
ал
Solve
=
=
.
Q₁ * =
Q₂² =

Expert Solution

Step 1: Writing the Lagrange function

Given $u open parentheses Q subscript 1 comma Q subscript 2 close parentheses equals Q subscript 1 superscript alpha Q subscript 2 superscript 1 minus alpha end superscript$ such that $y equals P subscript 1 Q subscript 1 plus P subscript 2 Q subscript 2.$

Consider the Lagrange function

$L open parentheses Q subscript 1 comma Q subscript 2 close parentheses equals u open parentheses Q subscript 1 comma Q subscript 2 close parentheses minus lambda open parentheses y minus P subscript 1 Q subscript 1 minus P subscript 2 Q subscript 2 close parentheses L open parentheses Q subscript 1 comma Q subscript 2 close parentheses equals Q subscript 1 superscript alpha Q subscript 2 superscript 1 minus alpha end superscript minus lambda open parentheses y minus P subscript 1 Q subscript 1 minus P subscript 2 Q subscript 2 close parentheses.$

We need to compute the derivatives and find the solution to the system of equations.

We will assume $P subscript 1 comma P subscript 2 comma alpha$ are constants and $lambda$ is a Lagrange multiplier.