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Question Number 70232 by Joel122 last updated on 02/Oct/19
$$\mathrm{Solve} \\ $$$${xy}'\:−\:{y}\:\mathrm{sin}\:{x}\:+\:{y}^{\mathrm{5}} \:=\:\mathrm{0} \\ $$
Commented by Joel122 last updated on 02/Oct/19
$$\mathrm{I}\:\mathrm{think}\:\mathrm{it}\:\mathrm{is}\:\mathrm{a}\:\mathrm{Bernoulli}'\mathrm{s}\:\mathrm{equation} \\ $$$${y}'\:−\:\left(\frac{\mathrm{sin}\:{x}}{{x}}\right){y}\:=\:−\frac{{y}^{\mathrm{5}} }{{x}} \\ $$$$\mathrm{using}\:{u}\:=\:{y}^{−\mathrm{4}} \:\mathrm{gives} \\ $$$${u}'\:+\:\left(\frac{\mathrm{4sin}\:{x}}{{x}}\right){u}\:=\:\frac{\mathrm{4}}{{x}} \\ $$$$\mathrm{and}\:\mathrm{integrating}\:\mathrm{factor}\:{F}\left({x}\right)\:=\:{e}^{\int\:\frac{\mathrm{4}\:\mathrm{sin}\:{x}}{{x}}\:{dx}} \\ $$
Commented by Joel122 last updated on 02/Oct/19
$$\mathrm{Now}\:\mathrm{I}'\mathrm{m}\:\mathrm{stuck}.\:\mathrm{Can}\:\mathrm{anyone}\:\mathrm{help}?\:\mathrm{Or} \\ $$$$\mathrm{is}\:\mathrm{it}\:\mathrm{any}\:\mathrm{other}\:\mathrm{way}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{this}? \\ $$
Commented by mind is power last updated on 02/Oct/19
$$\int\frac{{sin}\left({x}\right)}{{x}}={Si}\left({x}\right)\:{Sinus}\:{integral}\:{function} \\ $$$${But}\:{after}\:{that}\:{we}\:{cant}\:{find}\:{particular}\:{solution} \\ $$$${particular}\:{solution}\:{will}\:{bee}\:{by}\:{power}\:{Series} \\ $$