Question Number 40442 by rahul 19 last updated on 21/Jul/18
$$\mathrm{Solve}\:: \\ $$$$\mathrm{y}^{\mathrm{4}} \mathrm{d}{x}\:+\:\mathrm{2}{x}\mathrm{y}^{\mathrm{3}} \mathrm{dy}\:=\:\frac{\mathrm{yd}{x}−\:{x}\mathrm{dy}}{{x}^{\mathrm{3}} \mathrm{y}^{\mathrm{3}} }. \\ $$
Answered by ajfour last updated on 21/Jul/18
$${y}^{\mathrm{2}} \left({y}^{\mathrm{2}} {dx}+\mathrm{2}{xydy}\right)=\frac{{d}\left({x}/{y}\right)}{{x}^{\mathrm{3}} {y}} \\ $$$$\Rightarrow\:{y}^{\mathrm{2}} {d}\left({xy}^{\mathrm{2}} \right)=\frac{{d}\left({x}/{y}\right)}{{x}^{\mathrm{3}} {y}}\:\:\:\:\:\:…\left({i}\right) \\ $$$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ $$$$\Leftrightarrow\:\left({xy}^{\mathrm{2}} \right)^{{m}} {d}\left({xy}^{\mathrm{2}} \right)=\left(\frac{{x}}{{y}}\right)^{{n}} {d}\left(\frac{{x}}{{y}}\right) \\ $$$$\Rightarrow\:\:{m}−{n}=\mathrm{3}\:\:\:{and} \\ $$$$\:\:\:\:\:\:\:\mathrm{2}{m}+{n}=\mathrm{3} \\ $$$$\Rightarrow\:\:\:{m}=\mathrm{2}\:\:{and}\:\:{n}=−\mathrm{1} \\ $$$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ $$$${So}\:\:\:\left({i}\right)\:{becomes} \\ $$$$\:\:\:\:\int\left({xy}^{\mathrm{2}} \right)^{\mathrm{2}} {d}\left({xy}^{\mathrm{2}} \right)=\:\int\frac{{d}\left({x}/{y}\right)}{\left({x}/{y}\right)} \\ $$$$\Rightarrow\:\frac{\left({xy}^{\mathrm{2}} \right)^{\mathrm{3}} }{\mathrm{3}}=\mathrm{ln}\:\frac{{x}}{{y}}+{c}\:. \\ $$
Commented by rahul 19 last updated on 22/Jul/18
Great! ����
Answered by tanmay.chaudhury50@gmail.com last updated on 21/Jul/18
$$\:{y}^{\mathrm{2}} {dx}+\mathrm{2}{xydy}=\frac{\mathrm{1}}{{x}^{\mathrm{3}} {y}^{\mathrm{3}} }.{d}\left(\frac{{x}}{{y}}\right) \\ $$$${y}^{\mathrm{2}} {d}\left({x}\right)+{xd}\left({y}^{\mathrm{2}} \right)=\frac{\mathrm{1}}{{x}^{\mathrm{3}} {y}^{\mathrm{3}} }.{d}\left(\frac{{x}}{{y}}\right) \\ $$$${d}\left({xy}^{\mathrm{2}} \right)=\frac{{d}\left(\frac{{x}}{{y}}\right)}{{y}^{\mathrm{6}} \left(\frac{{x}}{{y}}\right)^{\mathrm{3}} } \\ $$$${y}^{\mathrm{6}} {d}\left({xy}^{\mathrm{2}} \right)=\frac{{d}\left(\frac{{x}}{{y}}\right)}{\left(\frac{{x}}{{y}}\right)^{\mathrm{3}} } \\ $$$${contd}….{wait}\:{pls}…{recheck}\:{the}\:{questionpls}… \\ $$$$\:\frac{{x}^{\mathrm{2}} }{{y}^{\mathrm{2}} }\:.{y}^{\mathrm{6}} .{d}\left({xy}^{\mathrm{2}} \right)=\frac{{d}\left(\frac{{x}}{{y}}\right)}{\left(\frac{{x}}{{y}}\right)} \\ $$$$\left({xy}^{\mathrm{2}} \right)^{\mathrm{2}} {d}\left({xy}^{\mathrm{2}} \right)=\frac{{d}\left(\frac{{x}}{{y}}\right)}{\left(\frac{{x}}{{y}}\right)} \\ $$$$\frac{\left({xy}^{\mathrm{2}} \right)^{\mathrm{3}} }{\mathrm{3}}={ln}\left(\frac{{x}}{{y}}\right)+{lnc} \\ $$
Commented by rahul 19 last updated on 22/Jul/18
thank you sir ��