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
Yes, you are on the right track. After rearranging the equation to have all the terms on one side, you have:
(1 + x)dy/dx - xy - x - x^2 = 0
The next step is to express dy/dx in terms of y and x. To do this, we can divide both sides of the equation by (1 + x):
(1 + x)dy/dx = xy + x + x^2
dy/dx = (xy + x + x^2) / (1 + x)
Now, we have a separable differential equation. To solve it, we can multiply both sides of the equation by dx:
dy = (xy + x + x^2) / (1 + x) * dx
Now, we can separate the variables by multiplying both sides by (1 + x) and dividing by (xy + x + x^2):
(1 + x) dy = (xy + x + x^2) dx / (xy + x + x^2)
Simplifying, we have:
(1 + x) dy = dx
Now, we can integrate both sides:
∫(1 + x) dy = ∫dx
Integrating, we get:
y + (x^2 / 2) + cx = x + k
Where c is the constant of integration and k is another constant. Rearranging, we have:
y = -x^2/2 + x + k - cx
So, the solution to the given differential equation is:
y = -x^2/2 + x + k - cx
where k and c are constants.