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Category: Differential Equation

x-dy-dx-y-y-3-cos-x-

Question Number 129334 by bramlexs22 last updated on 15/Jan/21 $$\:\mathrm{x}\:\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\mathrm{y}\:+\:\mathrm{y}^{\mathrm{3}} \mathrm{cos}\:\mathrm{x}\: \\ $$ Answered by liberty last updated on 15/Jan/21 $$\:\mathrm{let}\:\mathrm{y}\:=\:\mathrm{w}^{−\frac{\mathrm{1}}{\mathrm{2}}} \:\rightarrow\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{w}^{−\mathrm{3}/\mathrm{2}} \:\frac{\mathrm{dw}}{\mathrm{dx}} \\ $$$$\:\Rightarrow−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{xw}^{−\mathrm{3}/\mathrm{2}}…

y-x-dy-dx-a-y-2-dx-dy-

Question Number 129222 by bramlexs22 last updated on 14/Jan/21 $$\:\mathrm{y}−\mathrm{x}\:\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\mathrm{a}\left(\mathrm{y}^{\mathrm{2}} +\frac{\mathrm{dx}}{\mathrm{dy}}\:\right) \\ $$ Answered by bobhans last updated on 14/Jan/21 $$\:{y}−{xy}'\:=\:{ay}^{\mathrm{2}} +\frac{{a}}{{y}'} \\ $$$$\:{yy}'−{x}\left({y}'\right)^{\mathrm{2}} ={ay}^{\mathrm{2}}…