math differential equations
posted by manny on .
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

hello i am retada and i love u

retada i believe that was unecessary

(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!] 
ok thanks