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1. Show that the following are Lie groups: (i) x = ex, (ii) x = √√√x² +ε, (y + x)x (iii) x = y = y +ε, x y (iv) x y 1+ Ex 1+ Ey 2. Show that the following equations are invariant under the given Lie group (i) x²+e=1+xy, x = ex, y=ey, (ii) y + 2xy²+x²+2y² + 2x = 0, x = (iii) x²-y²-2xy sin = 0, x x = y: 1+ Ex 1+ εχ

1. Show that the following are Lie groups: (i) x = ex, (ii) x = √√√x² +ε, (y + x)x (iii) x = y = y +ε, x y (iv) x y 1+ Ex 1+ Ey 2. Show that the following equations are invariant under the given Lie group (i) x²+e=1+xy, x = ex, y=ey, (ii) y + 2xy²+x²+2y² + 2x = 0, x = (iii) x²-y²-2xy sin = 0, x x = y: 1+ Ex 1+ εχ

Advanced Engineering Mathematics
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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1. Show that the following are Lie groups: (i) x = ex, (ii) x = √√√x² +ε, (y + x)x (iii) x = y = y +ε, x y (iv) x y 1+ Ex 1+ Ey 2. Show that the following equations are invariant under the given Lie group (i) x²+e=1+xy, x = ex, y=ey, (ii) y + 2xy²+x²+2y² + 2x = 0, x = (iii) x²-y²-2xy sin = 0, x x = y: 1+ Ex 1+ εχ
Transcribed Image Text:1. Show that the following are Lie groups: (i) x = ex, (ii) x = √√√x² +ε, (y + x)x (iii) x = y = y +ε, x y (iv) x y 1+ Ex 1+ Ey 2. Show that the following equations are invariant under the given Lie group (i) x²+e=1+xy, x = ex, y=ey, (ii) y + 2xy²+x²+2y² + 2x = 0, x = (iii) x²-y²-2xy sin = 0, x x = y: 1+ Ex 1+ εχ
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