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Maths-

General

Easy

Question

The solution of the differential equation $left parenthesis sin invisible function application x plus cos invisible function application x right parenthesis d y plus left parenthesis cos invisible function application x minus sin invisible function application x right parenthesis d x equals 0$ is

$e^{x} (\sin x + \cos x) + c = 0$
$e^{y} (\sin x + \cos x) = c$
$e^{y} (\cos x - \sin x) = c$
$e^{x} (\sin x - \cos x) = c$

The correct answer is: $e to the power of y end exponent left parenthesis sin invisible function application x plus cos invisible function application x right parenthesis equals c$

$fraction numerator d y over denominator d x end fraction equals negative fraction numerator cos invisible function application x minus sin invisible function application x over denominator sin invisible function application x plus cos invisible function application x end fraction$ Þ $d y equals negative open parentheses fraction numerator cos invisible function application x minus sin invisible function application x over denominator sin invisible function application x plus cos invisible function application x end fraction close parentheses d x$
On integrating both sides, we get
Þ $y equals negative log invisible function application left parenthesis sin invisible function application x plus cos invisible function application x right parenthesis plus log invisible function application c$
Þ $y equals log invisible function application open parentheses fraction numerator c over denominator sin invisible function application x plus cos invisible function application x end fraction close parentheses$ Þ $e to the power of y end exponent left parenthesis sin invisible function application x plus cos invisible function application x right parenthesis equals c$ .

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