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$Solution of differential equation \frac{d z}{d x} + \frac{z}{x} \log z = \frac{z}{x^{2}} (\log z)^{2} is$

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a

xlogz=12+cx2

b

xlogz=12−cx2

c

xlogz=13+cx2

d

none of these

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detailed solution

Correct option is A

Dividing the complete equation by zlogz2 the transformed equation obtained is1z(log⁡z)2dzdx+1xlog⁡z=1x2; Now substituting 1log⁡z=y⇒−12(log⁡z)2dzdx=dydx………..(i) Then (i) reduced to linear differential equation as below −dydx+1xy=1x2⇒dydx−1xy=−1x2 and I.F=e−1∫1xdx=1x Its solution is given by yx=−∫1x3dx+c⇒yx=12x2+c⇒yx=12x2+c⇒xlog⁡z=12+cx2 is the required general solution

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