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Question Number 196843 by sonukgindia last updated on 01/Sep/23

Answered by qaz last updated on 02/Sep/23

$${xyy}^{″}={yy}^{\prime }+x-x{\left(y^{\prime }\right)}^2$$$$\Rightarrow x{\left({yy}^{\prime }\right)}^{\prime }={yy}^{\prime }+x$$$${yy}^{\prime }=e^{\int \frac{dx}{x}}(C_1+\int e^{-\int \frac{dx}{x}}dx)=C_{1}x+xlnx$$$$\Rightarrow y^2=C_1{x}^2+x^{2}lnx+C_2,C_1=\frac{1}{2},C_2=\frac{7}{2}$$$$y^2=\frac{1}{2}{x}^2+x^{2}lnx+\frac{7}{2}$$

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