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Question Number 91211 by john santu last updated on 28/Apr/20

$$xdy+5ydx=2y^{4}xdx$$

Commented by john santu last updated on 29/Apr/20

Answered by MWSuSon last updated on 28/Apr/20

$$x\frac{dy}{dx}+5y=2y^{4}x$$$$y^{-4}\frac{dy}{dx}+\frac{5y^{-3}}{x}=2$$$$letv=y^{-3}$$$$\frac{dv}{dx}=-3y^{-4}\frac{dy}{dx}$$$$\frac{dv}{dx}-\frac{15v}{x}=-6$$$$I.F=exp(-15\int \frac{1}{x}dx)=exp(-15{log}_{e}x)$$$$x>0$$$$x^{-15}\frac{dv}{dx}-15x^{-16}v=6x^{-15}$$$$\frac{d}{dx}\left(x^{-15}v\right)=6x^{-15}$$$$x^{-15}v=\int 6x^{-15}dx$$$$x^{-15}{y}^{-3}=\frac{3x^{-14}}{7}+c$$$$y{=}^3\sqrt{\frac{x^{-15}}{\frac{3x^{-14}}{7}+c}}$$

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