Question and Answers Forum

All Questions      Topic List

Differential Equation Questions

Previous in All Question      Next in All Question      

Previous in Differential Equation      Next in Differential Equation      

Question Number 129126 by benjo_mathlover last updated on 13/Jan/21

(dy/dx)−y = xy^5

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{dy}}{\mathrm{dx}}−\mathrm{y}\:=\:\mathrm{xy}^{\mathrm{5}} \: \\ $$

Answered by liberty last updated on 13/Jan/21

let v = y^(−4) ; (dv/dx) =−4y^(−5 ) (dy/dx) (dy/dx) = −(1/4)y^5 (dv/dx) (substitute into diff eq ) give −(1/4)y^5 (dv/dx)−y = xy^5 (dv/dx) + 4v = −4x ; put integrating factor μ =e^(∫ 4 dx) = e^(4x) , then we get v = ((∫ −4x.e^(4x) dx + C)/e^(4x) ) (1/y^4 ) = ((−xe^(4x) +(1/4)e^(4x) +C)/e^(4x) ) y^4 = (e^(4x) /(e^(4x) (−x+(1/4))+C))

$$\:\mathrm{let}\:\mathrm{v}\:=\:\mathrm{y}^{−\mathrm{4}} \:;\:\frac{\mathrm{dv}}{\mathrm{dx}}\:=−\mathrm{4y}^{−\mathrm{5}\:} \frac{\mathrm{dy}}{\mathrm{dx}} \\ $$$$\:\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:−\frac{\mathrm{1}}{\mathrm{4}}\mathrm{y}^{\mathrm{5}} \:\frac{\mathrm{dv}}{\mathrm{dx}}\:\left(\mathrm{substitute}\:\mathrm{into}\:\mathrm{diff}\:\mathrm{eq}\:\right) \\ $$$$\mathrm{give}\:−\frac{\mathrm{1}}{\mathrm{4}}\mathrm{y}^{\mathrm{5}} \:\frac{\mathrm{dv}}{\mathrm{dx}}−\mathrm{y}\:=\:\mathrm{xy}^{\mathrm{5}} \\ $$$$\:\frac{\mathrm{dv}}{\mathrm{dx}}\:+\:\mathrm{4v}\:=\:−\mathrm{4x}\:;\:\mathrm{put}\:\mathrm{integrating}\:\mathrm{factor} \\ $$$$\:\mu\:=\mathrm{e}^{\int\:\mathrm{4}\:\mathrm{dx}} =\:\mathrm{e}^{\mathrm{4x}} \:,\:\mathrm{then}\:\mathrm{we}\:\mathrm{get}\: \\ $$$$\:\mathrm{v}\:=\:\frac{\int\:−\mathrm{4x}.\mathrm{e}^{\mathrm{4x}} \:\mathrm{dx}\:+\:\mathrm{C}}{\mathrm{e}^{\mathrm{4x}} } \\ $$$$\:\frac{\mathrm{1}}{\mathrm{y}^{\mathrm{4}} }\:=\:\frac{−\mathrm{xe}^{\mathrm{4x}} +\frac{\mathrm{1}}{\mathrm{4}}\mathrm{e}^{\mathrm{4x}} \:+\mathrm{C}}{\mathrm{e}^{\mathrm{4x}} } \\ $$$$\underline{\:}\boldsymbol{\mathrm{y}}^{\mathrm{4}} \:=\:\frac{\boldsymbol{\mathrm{e}}^{\mathrm{4}\boldsymbol{\mathrm{x}}} }{\boldsymbol{\mathrm{e}}^{\mathrm{4}\boldsymbol{\mathrm{x}}} \left(−\boldsymbol{\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{4}}\right)+\boldsymbol{\mathrm{C}}}\underline{\:}\:\: \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com