Solve the differential equation `(dy)/(dx) + xy = xy^2` for `y` in terms of `x` given `y=3` when `x=0`

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We have the differential equation `(dy)/(dx) +xy = xy^2`

and that `y=3` when `x=0`

Rearranging the differential equation we get

`(dy)/(dx) = x(y^2-y)`

`implies`  `(1/(y^2-y))(dy)/(dx) =x`   (terms on the left are terms in `y` and `(dy)/(dx)`, terms on the right are terms in `x`)


Now, integrate both sides with respect to `x` giving

`int (1/(y(y-1))) (dy)/(dx) dx = int x dx`

`implies` `int(1/(y(y-1))) dy = x^2/2 + c`    (where `c` is a constant)

`implies int -1/y dy + int1/(y-1) dy = x^2/2 + c`

`implies`  `-ln(y) + ln(y-1) = x^2/2 + k`  (where `k`is a constant)

`implies`  `ln((y-1)/y) = x^2/2+k`

`implies`  `(y-1)/y = e^(x^2/2)e^k`

Now, we have that `y=3`  when `x=0`

Substituting these values in we get

`2/3 = e^0e^k = e^k` 

Therefore   `(y-1)/y = 2/3e^(x^2/2)`

`implies`  `y = (2/3e^(x^2/2))y +1`

`implies`  `y(1-2/3e^(x^2/2)) = 1`

`implies`  `y = 1/(1-2/3e^(x^2/2))`       solution

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