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Question Number 70232 by Joel122 last updated on 02/Oct/19

$$\mathrm{Solve}$$$${xy}^{\prime }-y\sin x+y^5=0$$

Commented by Joel122 last updated on 02/Oct/19

$$\mathrm{I}\mathrm{think}\mathrm{it}\mathrm{is}\mathrm{a}{\mathrm{Bernoulli}}^{\prime }\mathrm{s}\mathrm{equation}$$$$y^{\prime }-\left(\frac{\sin x}{x}\right)y=-\frac{y^5}{x}$$$$\mathrm{using}u=y^{-4}\mathrm{gives}$$$$u^{\prime }+\left(\frac{4\sin x}{x}\right)u=\frac{4}{x}$$$$\mathrm{and}\mathrm{integrating}\mathrm{factor}F\left(x\right)=e^{\int \frac{4\sin x}{x}dx}$$

Commented by Joel122 last updated on 02/Oct/19

$$\mathrm{Now}{\mathrm{I}}^{\prime }\mathrm{m}\mathrm{stuck}.\mathrm{Can}\mathrm{anyone}\mathrm{help}?\mathrm{Or}$$$$\mathrm{is}\mathrm{it}\mathrm{any}\mathrm{other}\mathrm{way}\mathrm{to}\mathrm{solve}\mathrm{this}?$$

Commented by mind is power last updated on 02/Oct/19

$$\int \frac{sin\left(x\right)}{x}=Si\left(x\right)Sinusintegralfunction$$$$Butafterthatwecantfindparticularsolution$$$$particularsolutionwillbeebypowerSeries$$

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