Nonhomogeneous 2nd order dif equation

[darjiaus7]

Hello guys . I want to do nonhomogeneous 2nd order dif equation..I am trying to do this for 2 days , but I can't get good answer. Can you show me how to do this equation with constant variation method ( i know it best) or other I would be very gratefull , because after 2 days of trying I am surrendered... $xy''-2y' = -\frac{2}{x^2} ; y(1)=-\frac{1}{3} ; y'(1)=1$

 

[pasmith]

Hello guys . I want to do nonhomogeneous 2nd order dif equation..I am trying to do this for 2 days , but I can't get good answer. Can you show me how to do this equation with constant variation method ( i know it best) or other I would be very gratefull , because after 2 days of trying I am surrendered... $xy''-2y' = -\frac{2}{x^2} ; y(1)=-\frac{1}{3} ; y'(1)=1$

This is a first-order ODE for $u = y'$: $$u' -\frac{2}{x}u = -\frac{2}{x^3}.$$ It has an integrating factor.

 

[lurflurf]

change variables
$\mathrm{u}(x)=x^{-2} \mathrm{y}^\prime (x)$