1. For the given Lie group, find the corresponding infinitesimals X and Y. (i) x=coshex+sinhey, y=coshey+sinh Ex (ii) =√√x²-2y+2yeε, y = yeε, (iii) x=x √1+2E12 y = y √1+2ɛy2 2. For the given infinitesimals X and Y, find the corresponding Lie group. (i) x=xy, Y=1 (ii) X=x+y, Y = y, (iii) X = x²-x, Y = y² + y. 3. Find infinitesimal transformations leaving the following ODEs invari- ant. Use these to find a change of variables and reduce the original ODE to one that is separable and solve the equation. (i) (ii) ** ** dy = xy + dx x xy al× dy (iii) = dx dy dx = 3y + 2y+x3 y+y³ x+(x+1)y² Hints: Try: 1. X = 4(x), Y = B(x)y, 2. X=A(x), Y = B(y), 3. X = 4(y), Y = B(x).

1. For the given Lie group, find the corresponding infinitesimals X and Y. (i) x=coshex+sinhey, y=coshey+sinh Ex (ii) =√√x²-2y+2yeε, y = yeε, (iii) x=x √1+2E12 y = y √1+2ɛy2 2. For the given infinitesimals X and Y, find the corresponding Lie group. (i) x=xy, Y=1 (ii) X=x+y, Y = y, (iii) X = x²-x, Y = y² + y. 3. Find infinitesimal transformations leaving the following ODEs invari- ant. Use these to find a change of variables and reduce the original ODE to one that is separable and solve the equation. (i) (ii) dy = xy + dx x xy al× dy (iii) = dx dy dx = 3y + 2y+x3 y+y³ x+(x+1)y² Hints: Try: 1. X = 4(x), Y = B(x)y, 2. X=A(x), Y = B(y), 3. X = 4(y), Y = B(x).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.3: Lines

Problem 14E

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