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solve-xy-2y-x-4-




Question Number 30187 by abdo imad last updated on 17/Feb/18
solve xy^′  −2y = x^4  .
$${solve}\:{xy}^{'} \:−\mathrm{2}{y}\:=\:{x}^{\mathrm{4}} \:. \\ $$
Answered by ajfour last updated on 18/Feb/18
(dy/dx)−((2/x))y=x^3   e^(−∫(2/x)dx)  = e^(−2ln x)  = (1/x^2 )   ⇒  (y/x^2 ) = ∫xdx +c_1   or    (y/x^2 ) = (x^2 /2) +c  or          y = (x^4 /2)+cx^2    .
$$\frac{{dy}}{{dx}}−\left(\frac{\mathrm{2}}{{x}}\right){y}={x}^{\mathrm{3}} \\ $$$${e}^{−\int\frac{\mathrm{2}}{{x}}{dx}} \:=\:{e}^{−\mathrm{2ln}\:{x}} \:=\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} } \\ $$$$\:\Rightarrow\:\:\frac{{y}}{{x}^{\mathrm{2}} }\:=\:\int{xdx}\:+{c}_{\mathrm{1}} \\ $$$${or}\:\:\:\:\frac{{y}}{{x}^{\mathrm{2}} }\:=\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\:+{c} \\ $$$${or}\:\:\:\:\:\:\:\:\:\:\boldsymbol{{y}}\:=\:\frac{\boldsymbol{{x}}^{\mathrm{4}} }{\mathrm{2}}+\boldsymbol{{cx}}^{\mathrm{2}} \:\:\:. \\ $$

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