Find a function y of x such that 2yy′=x and y(2)=6

Answer:

$y=\sqrt{(x^(2))/(2) +34}$

Explanation:

$2y\left( x\right) y^(\prime )\left( x\right) =x$

$\Longrightarrow \int 2y\left( x\right) y^(\prime )\left( x\right) dx=\int x\ dx$

$\Longrightarrow y^(2)\left( x\right) =(x^(2))/(2) +c\ \text{(c is a constant)}$

$y\left( 2\right) =6\Longleftrightarrow y^(2)\left( 2\right) =36$

$y^(2)\left( 2\right) =36\Longleftrightarrow (\left( 2\right)^(2) )/(2) +c=36\ \Longleftrightarrow c=34$

$\Longrightarrow y\left( x\right) =\pm\sqrt{(x^(2))/(2) +34}$

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