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CBSE Class 12-science Answered

Differential equation. 
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Asked by alanpeter9611 | 15 Feb, 2019, 06:27: PM
answered-by-expert Expert Answer
Only one question should be asked per posting. Qn.1 is answered below
 
Given differential equation can be written as
 
begin mathsize 12px style fraction numerator d y over denominator d x end fraction equals space fraction numerator 2 x space sin y over x space minus space y space cos begin display style y over x end style over denominator y minus x space cos space begin display style y over x end style end fraction space space space...................... left parenthesis 1 right parenthesis
p u t space space space y space equals space v space x space comma space space space h e n c e space fraction numerator d y over denominator d x end fraction space equals space v space plus space x space fraction numerator d v over denominator d x end fraction space space space space space................ left parenthesis 2 right parenthesis

B y space u sin g space e q n. left parenthesis 2 right parenthesis space comma space e q n. left parenthesis 1 right parenthesis space c a n space b e space r e w r i t t e n space a s

v space plus space x space fraction numerator d v over denominator d x end fraction space equals space fraction numerator 2 space sin space v space minus space v space cos space v over denominator v space minus space cos space v end fraction.............. left parenthesis 3 right parenthesis
space x space fraction numerator begin display style d v end style over denominator begin display style d x end style end fraction space equals space fraction numerator begin display style 2 space sin space v space minus space v space cos space v end style over denominator begin display style v space minus space cos space v end style end fraction space minus space v space equals space space fraction numerator 2 space sin space v space minus space v squared over denominator v space minus space cos space v end fraction
fraction numerator v space minus space cos space v over denominator 2 space sin space v space minus space v squared end fraction space d v space equals space fraction numerator d x over denominator x end fraction space space................. left parenthesis 4 right parenthesis

l e t space space u space equals space open parentheses 2 space sin space v space minus space v squared close parentheses space space space comma space space space space h e n c e space space space d u space equals space 2 open parentheses space cos space v space minus space v close parentheses space d v
h e n c e space e q n. left parenthesis 4 right parenthesis space i s space r e w r i t t e n space a s space space minus 1 half fraction numerator d u over denominator u end fraction space equals fraction numerator d x over denominator x end fraction space........... left parenthesis 5 right parenthesis
b y space i n t e g r a t i n g space e q n. left parenthesis 5 right parenthesis comma space w e space g e t comma space space space space space log space left parenthesis x right parenthesis space plus log space square root of u space equals space log space C

H e n c e space space w e space g e t comma space space space space space space x square root of u space equals space x space square root of 2 space sin space v space minus space v squared end root space space equals space x square root of 2 space sin open parentheses y over x close parentheses space minus space open parentheses fraction numerator begin display style y end style over denominator begin display style x end style end fraction close parentheses squared end root space equals space C
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Answered by Thiyagarajan K | 16 Feb, 2019, 04:32: PM
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