find the solution of dy/dx=sin(x+y)+cos(x+y)
 

Asked by Ritika Agrawal | 9th Oct, 2014, 10:31: PM

Expert Answer:

begin mathsize 14px style fraction numerator d y over denominator d x end fraction equals sin open parentheses x plus y close parentheses plus c o s open parentheses x plus y close parentheses text Substitute end text space x plus y equals v d i f f e r t i a t i n g space w. r. t. space x rightwards double arrow 1 plus fraction numerator d y over denominator d x end fraction equals fraction numerator d v over denominator d x end fraction rightwards double arrow fraction numerator d y over denominator d x end fraction equals fraction numerator d v over denominator d x end fraction minus 1  text The   given   differential   equation   becomes end text fraction numerator d y over denominator d x end fraction equals sin open parentheses x plus y close parentheses plus c o s open parentheses x plus y close parentheses rightwards double arrow fraction numerator d v over denominator d x end fraction minus 1 equals sin v plus cos v rightwards double arrow fraction numerator d v over denominator d x end fraction equals 1 plus sin v plus cos v rightwards double arrow fraction numerator 1 over denominator 1 plus sin v plus cos v end fraction d v equals d x rightwards double arrow fraction numerator 1 over denominator 1 plus begin display style fraction numerator 2 tan open parentheses v divided by 2 close parentheses over denominator 1 plus tan squared open parentheses v divided by 2 close parentheses end fraction end style plus fraction numerator 1 minus tan squared open parentheses v divided by 2 close parentheses over denominator 1 plus tan squared open parentheses v divided by 2 close parentheses end fraction end fraction d v equals d x rightwards double arrow fraction numerator 1 plus tan squared open parentheses v divided by 2 close parentheses over denominator 2 open parentheses 1 plus tan open parentheses v divided by 2 close parentheses close parentheses end fraction d v equals d x rightwards double arrow fraction numerator s e c squared open parentheses v divided by 2 close parentheses over denominator 2 open parentheses 1 plus tan open parentheses v divided by 2 close parentheses close parentheses end fraction d v equals d x rightwards double arrow integral fraction numerator s e c squared open parentheses v divided by 2 close parentheses over denominator 2 open parentheses 1 plus tan open parentheses v divided by 2 close parentheses close parentheses end fraction d v equals integral d x thin space thin space space space space text   [ integrating ] end text rightwards double arrow log open vertical bar 2 left parenthesis 1 plus tan open parentheses v divided by 2 close parentheses right parenthesis close vertical bar plus c equals x text Resubstituting end text space x plus y equals v rightwards double arrow x equals log open vertical bar 2 left parenthesis 1 plus tan open parentheses fraction numerator x plus y over denominator 2 end fraction close parentheses close vertical bar plus c space space space space space space space space left parenthesis c space i s space c o n s tan t space o f space i n t e g r a t i o n right parenthesis     end style

Answered by Vijaykumar Wani | 10th Oct, 2014, 10:53: AM

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