Answered: x₁ + x₂ + x3 + x4 + x5 < 81, all x > 0.

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

How many integer solutions? Please provide a complete justification and answer written clearly.

C.
x₁ + x₂ + x3 + x4 + X5 < 81, all x₂ > 0.

Expert Solution

Step 1: describing the given conditions

given

$x_{1} + x_{2} + x_{3} + x_{4} + x_{5} < 81 . . . . . . . . . . . . (1) a n d a l l x_{i} > 0$

now we add one integer x₆ $greater than$ 0 such that

$x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} = 81 . . . . . . . . . . . . (2)$

$space W e space r e d u c e space t h e space p r o b l e m space t o space o n e space i n space t h e space n o n minus n e g a t i v e space i n t e g e r s. space L e t space y subscript k equals x subscript k minus 1 space space f o r space 1 less or equal than k less or equal than 5 space. space T h e n space e a c h space y subscript k space space i s space a space n o n minus n e g a t i v e space i n t e g e r. space S u b s t i t u t i n g space y subscript k plus 1 space space f o r space x subscript k space comma space 1 less or equal than k less or equal than 5 space comma space i n space e q u a t i o n space left parenthesis 2 right parenthesis space w e space g e t$

$y_{1} + 1 + y_{2} + 1 + y_{3} + 1 + y_{4} + 1 + y_{5} + 1 + y_{6} + 1 = 81 . . . . . . . . . . . . (2)$