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Question Number 67653 by mhmd last updated on 29/Aug/19

Answered by mr W last updated on 29/Aug/19

$$lett=\frac{y}{x}$$$$y=xt$$$$y^{\prime }=t+{xt}^{\prime }$$$$$$$${xy}^{\prime }=y+2x^3{\sin }^2\left(\frac{y}{x}\right)$$$$y^{\prime }=\frac{y}{x}+2x^2{\sin }^2\left(\frac{y}{x}\right)$$$$t+{xt}^{\prime }=t+2x^2{\sin }^{2}t$$$$t^{\prime }=2x{\sin }^{2}t$$$$\frac{dt}{{\sin }^{2}t}=2xdx$$$$\int \frac{dt}{{\sin }^{2}t}=\int 2xdx$$$$-\frac{1}{\tan t}=x^2+C$$$$\Rightarrow (x^2+C)\tan \frac{y}{x}+1=0$$$$or$$$$\Rightarrow y=-x{\tan }^{-1}\frac{1}{x^2+C}$$

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