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Solution of diffrenrtial equation $\frac{dy}{dx} = \frac{y}{ylny-x}$ is

Option 1)
$4xy+2y^2lny - y^2=C$

Option 2)
$4xy+2y^2lny + y^2=C$

Option 3)
$4xy-2y^2lny - y^2=C$

Option 4)
$4xy-2y^2lny + y^2=C$

Answers (1)

best_answer

As we have learned

Linear Differential Equation -

Multiply by $e^{SPdx}$ which is the Integrating factor

- wherein

P is the function of x alone

Like $\frac{dy}{dx}+ Py = Q$ , where P, Q are function of x alone $\frac{dx}{dy}+ Px = Q$ , Where P,Q

are function of y alone is also a form of linear diffrential equation . So given equation here can

be written as $\frac{dx}{dy}= \frac{ylny-x}{y}$

$\frac{dx}{dy}+ x/y= lny$ on compairing it with $\frac{dx}{dy}+ px= Q , we get p= 1/y and Q = lny$

integrating factor $e^{1/ydy}= e^{lny }= y$

multiplying both sides by integrating factor , we get

$y \frac{dy}{dx}+ x = y lny$

$\Rightarrow \frac{d}{dy}(yx)= y lny$

$\Rightarrow d(xy)- y lnydy= 0$

$\Rightarrow\int d(xy)- \int y lnydy= c$

using integration by parts , we get ,

$xy - \frac{y^2}{2}lny + \frac{y^2}{4}= C$

$\Rightarrow 4xy-2y^lny + y^2=C$

Option 1)

$4xy+2y^2lny - y^2=C$

Option 2)

$4xy+2y^2lny + y^2=C$

Option 3)

$4xy-2y^2lny - y^2=C$

Option 4)

$4xy-2y^2lny + y^2=C$

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