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Question

The solution of the differential equation $fraction numerator d y over denominator d x end fraction equals fraction numerator x y over denominator x to the power of 2 end exponent plus y to the power of 2 end exponent end fraction$ is

$a y^{2} = e^{x^{2} / y^{2}}$
$a y = e^{x / y}$
$y = e^{x^{2}} + e^{y^{2}} + c$
$y = e^{x^{2}} + y^{2} + c$

The correct answer is: $a y to the power of 2 end exponent equals e to the power of x to the power of 2 end exponent divided by y to the power of 2 end exponent end exponent$

Given $fraction numerator d y over denominator d x end fraction equals fraction numerator x y over denominator x to the power of 2 end exponent plus y to the power of 2 end exponent end fraction$ . Put $y equals v x$ ; $fraction numerator d y over denominator d x end fraction equals v plus x. fraction numerator d v over denominator d x end fraction$
$therefore v plus x fraction numerator d v over denominator d x end fraction equals fraction numerator left parenthesis x right parenthesis left parenthesis v x right parenthesis over denominator x to the power of 2 end exponent plus v to the power of 2 end exponent x to the power of 2 end exponent end fraction$
Þ $v plus x. fraction numerator d v over denominator d x end fraction equals fraction numerator v over denominator 1 plus v to the power of 2 end exponent end fraction$ Þ $x fraction numerator d v over denominator d x end fraction equals fraction numerator negative v to the power of 3 end exponent over denominator 1 plus v to the power of 2 end exponent end fraction$
Þ $fraction numerator left parenthesis 1 plus v to the power of 2 end exponent right parenthesis over denominator v to the power of 3 end exponent end fraction d v equals negative fraction numerator d x over denominator x end fraction$ Þ $open parentheses fraction numerator 1 over denominator v to the power of 3 end exponent end fraction plus fraction numerator 1 over denominator v end fraction close parentheses d v equals negative fraction numerator d x over denominator x end fraction$
Integrating both sides, $not stretchy integral fraction numerator d v over denominator v to the power of 3 end exponent end fraction plus not stretchy integral fraction numerator d v over denominator v end fraction equals negative not stretchy integral fraction numerator d x over denominator x end fraction$
Þ $negative fraction numerator 1 over denominator 2 v to the power of 2 end exponent end fraction plus log invisible function application v equals negative log invisible function application x minus log invisible function application c$
Þ $negative fraction numerator x to the power of 2 end exponent over denominator 2 y to the power of 2 end exponent end fraction plus log invisible function application y equals negative log invisible function application c$ Þ $log invisible function application c y equals fraction numerator x to the power of 2 end exponent over denominator 2 y to the power of 2 end exponent end fraction$
Þ $c y equals e to the power of x to the power of 2 end exponent divided by 2 y to the power of 2 end exponent end exponent$ Þ $c to the power of 2 end exponent y to the power of 2 end exponent equals e to the power of x to the power of 2 end exponent divided by y to the power of 2 end exponent end exponent$
$therefore y to the power of 2 end exponent a equals e to the power of x to the power of 2 end exponent divided by y to the power of 2 end exponent end exponent$ , where $c to the power of 2 end exponent equals a$ .

Related Questions to study

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parallel

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