question_answer

Solve the differential equation \[(x\,dy-y\,dx)\,\,y\,\,\sin \left( \frac{y}{x} \right)\] \[-(y\,\,dx+x\,\,dy)\,x\,\,\cos \left( \frac{y}{x} \right)=0.\]

Answer:

We have, \[(xdy-ydx)\,y\sin \,\left( \frac{y}{x} \right)=(ydx+xdy)\,x\cos \left( \frac{y}{x} \right)\] \[\Rightarrow \]   \[xy\sin \left( \frac{y}{x} \right)dy-{{y}^{2}}\sin \left( \frac{y}{x} \right)\] \[\Rightarrow \]   \[dx=xy\cos \left( \frac{y}{x} \right)-{{x}^{2}}{{\cos }^{2}}\left( \frac{y}{x} \right)\,dy\] \[\Rightarrow \]   \[\left[ xy\sin \,\left( \frac{y}{x} \right)-{{x}^{2}}\cos \,\left( \frac{y}{x} \right) \right]dy\] \[=\left[ xy\cos \,\left( \frac{y}{x} \right)-{{y}^{2}}\sin \,\left( \frac{y}{x} \right) \right]dy\] \[\Rightarrow \]   \[\frac{dy}{dx}=\frac{xy\cos \,(y/x)+{{y}^{2}}\sin \,(y/x)}{xy\sin \,(y/x)-{{x}^{2}}\cos \,(y/x)}\] Dividing numerator and denominator by \[{{x}^{2}},\]we get \[\frac{dy}{du}=\frac{y/x\cos \left( \frac{y}{x} \right)+{{\left( \frac{y}{x} \right)}^{2}}\cos \left( \frac{y}{x} \right)}{y/x\sin \,(y/x)-\cos \left( \frac{y}{x} \right)}\]  ?(i) Put       \[y=vx\Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}\] Putting the values of \[\frac{dy}{dx}\]and y in Eq. (i), we get \[\Rightarrow \]\[v+x\frac{dv}{dx}=\frac{\frac{vx}{x}\cos \left( \frac{vx}{x} \right)+\frac{{{v}^{2}}{{x}^{2}}}{{{x}^{2}}}\sin \left( \frac{vx}{x} \right)}{\frac{vx}{x}\sin \left( \frac{vx}{x} \right)-\cos \left( \frac{vx}{x} \right)}\] \[\Rightarrow \]   \[v+x\frac{dv}{dx}=\frac{v\cos v+{{v}^{2}}\sin v}{v\sin v-\cos v}\] \[\Rightarrow \]\[x\frac{dv}{dx}=\frac{v\cos v+{{v}^{2}}\sin v-v\,(v\sin v-\cos v)}{v\sin v-\cos v}\] \[\Rightarrow \]\[x\frac{dv}{dx}=\frac{v\cos v+{{v}^{2}}\sin v-{{v}^{2}}\sin v-v\cos v}{v\sin v-\cos v}\] \[\Rightarrow \]   \[\left( \frac{v\sin v-\cos v}{v\cos v} \right)\,dv=2\frac{dx}{x}\] \[\Rightarrow \]   \[\left( \frac{v\sin v}{v\cos v}-\frac{\cos v}{v\cos v} \right)\,dv=2\frac{dx}{x}\] \[\Rightarrow \]   \[\left( \tan v-\frac{1}{v} \right)\,dv=2\frac{dx}{x}\] On integrating both sides, we get \[\Rightarrow \]   \[\int{\left( \tan v-\frac{1}{v} \right)\,dv=2\int{\frac{dx}{x}}}\] \[\Rightarrow \]   \[\int{\tan \,\,vdv-\int{\frac{dv}{v}}-2\int{\frac{dx}{x}}}\] \[\Rightarrow \]   \[\log |\sec v|-\,\log |v|\,\,=2\log |x|+\log |C|\] \[\Rightarrow \]   \[\log \left| \frac{\sec v}{v} \right|\,\,=\log |{{x}^{2}}C|\,\,\Rightarrow \frac{\sec v}{v}={{x}^{2}}C\] \[\Rightarrow \]   \[\frac{\sec \,(y/x)}{(y/x)}={{x}^{2}}C\]                  \[\left[ \because v=\frac{y}{x} \right]\] \[\Rightarrow \]   \[\sec \,(y/x)=(y/x){{x}^{2}}C\] \[\Rightarrow \]   \[\sec \left( \frac{y}{x} \right)=Cxy\]