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Question Number 77537 by john santu last updated on 07/Jan/20

$$\mathrm{what}\mathrm{is}\mathrm{the}\mathrm{solution}$$$${\mathrm{xy}}^{\prime }+(1+\mathrm{xcot}\mathrm{x})\mathrm{y}=\mathrm{x}?$$

Answered by mr W last updated on 07/Jan/20

$$y^{\prime }+(\frac{1}{x}+\frac{\cos x}{\sin x})y=1$$$$type{y}^{\prime }+p\left(x\right)y=q\left(x\right)$$$$\int p\left(x\right)dx=\int (\frac{1}{x}+\frac{\cos x}{\sin x})dx=\ln \left(x\right)+\ln \left(\sin x\right)=\ln \left(x\sin x\right)$$$$u\left(x\right)=e^{\int p\left(x\right)dx}=e^{\ln \left(x\sin x\right)}=x\sin x$$$$\int q\left(x\right)u\left(x\right)dx=\int x\sin xdx=\int xd(-\cos x)=-x\cos x+\int \cos xdx=-x\cos x+\sin x$$$$y=\frac{\int q\left(x\right)u\left(x\right)dx+C}{u\left(x\right)}=\frac{\sin x-x\cos x+C}{x\sin x}$$

Commented by mr W last updated on 07/Jan/20

Commented by john santu last updated on 08/Jan/20

$$\mathrm{thanks}\mathrm{you}\mathrm{sir}$$

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