math- differential equations

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This is what im given
(1 + x)dy/dx - xy = x + x2
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From my understanding I should move everything to one side like:

(1 + x)dy/dx -xy - x -x2 = 0

then dy/ dx = xy + x + x2/(1 + x)

and from here i am lost

Can you guys help!

Thanks

  • math- differential equations -

    hello i am retada and i love u

  • please answer! -

    retada i believe that was unecessary

  • math- differential equations -

    (1 + x)dy/dx - xy = x + x^2 --->

    y' - x/(1+x) y = x

    First solve the homogeneous part of the equation:

    y_h' - x/(1+x) y_h =0 --->

    y_h = A exp(x)/(1+x)

    Then you promote A from a constant to a function of x and put:

    y = A(x) Exp(x)/(1+x)

    If you subsitute this in the differential equation, then only the term involving the derivative of A will survive. That's because the terms that do not involve the derivative of A are exactly what you get when you take y to be y_h. y_h satisfies the homogeneous equation and therefore they add up to zero.

    So, we get:

    A' Exp(x)/(1+x) = x --->

    A' = x(1+x)Exp(-x)

    Integral of Exp(ax) = 1/a Exp(ax)

    Differentiate both sides w.r.t. a:

    Integral of x Exp(ax) =

    (x/a -1/a^2) Exp(ax)

    Differentiate again:

    Integral of x^2 Exp(ax) =

    ( x^2/a -2x/a^2 +2/a^3) Exp(ax)


    We thus see that:

    A(x) = (-3x -3 -x^2)Exp(-x) + const.

    [Note: I didn't check if I made any errors in the calculations!]

  • math- differential equations -

    ok thanks

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