the solution

Asked by reddyanji405 | 5th Jun, 2020, 10:29: AM

Expert Answer:

To find the solution of the differential equation: (2x - 3y + 5)dx + (9y - 6x -7)dy = 0
This can be done as follows:

open parentheses 2 straight x minus 3 straight y plus 5 close parentheses dx plus open parentheses 9 straight y minus 6 straight x minus 7 close parentheses dy equals 0
rightwards double arrow dy over dx equals fraction numerator negative open parentheses 2 straight x minus 3 straight y plus 5 close parentheses over denominator 9 straight y minus 6 straight x minus 7 end fraction
rightwards double arrow dy over dx equals fraction numerator open parentheses 2 straight x minus 3 straight y plus 5 close parentheses over denominator 3 open parentheses 2 straight x minus 3 straight y close parentheses plus 7 end fraction space space space... space space open parentheses straight i close parentheses
Let space straight u equals 2 straight x minus 3 straight y
rightwards double arrow du over dx equals 2 minus 3 dy over dx
rightwards double arrow du over dx equals 2 minus 3 open parentheses fraction numerator straight u plus 5 over denominator 3 straight u plus 7 end fraction close parentheses space space space space space... space space space From space left parenthesis straight i right parenthesis
rightwards double arrow du over dx equals fraction numerator 3 straight u minus 1 over denominator 3 straight u plus 7 end fraction
rightwards double arrow dx equals fraction numerator 3 straight u plus 7 over denominator 3 straight u minus 1 end fraction du
rightwards double arrow integral dx equals integral fraction numerator 3 straight u minus 1 plus 8 over denominator 3 straight u minus 1 end fraction du equals integral open parentheses 1 plus fraction numerator 8 over denominator 3 straight u minus 1 end fraction close parentheses du
rightwards double arrow straight x equals straight u plus 8 over 3 log open parentheses 3 straight u minus 1 close parentheses
rightwards double arrow 3 straight x equals 3 open parentheses 2 straight x minus 3 straight y close parentheses plus 8 log open vertical bar 3 left parenthesis 2 straight x minus 3 straight y right parenthesis minus 1 close vertical bar
rightwards double arrow 3 straight x minus 9 straight y plus 8 log open vertical bar 6 straight x minus 9 straight y minus 1 close vertical bar
Hence comma space option space left parenthesis straight b right parenthesis space is space correct.

Answered by Renu Varma | 5th Jun, 2020, 12:58: PM