Question

Solve the following differential equation `y'= (2x)/((x-1)^2(x+2))`?

Answer 1

`y = 4/9 ln|x-1| - 2/3 \ 1/(x-1) -4/9 ln|x+2| + C`

Explanation:

We seek a solution of the ODE:

`y' = (2x)/( (x-1)^2(x+2))`

This is a First Order Separable oDE, so we can "separate the variables" giving:

`int \ dy = int \ (2x)/( (x-1)^2(x+2)) \ dx` ..... [A]

The LHS integral is trivial, and for the RHS integral we can use a partial fraction decomposition of the integrand:

`(2x)/( (x-1)^2(x+2)) -= A/(x-1) + B/(x-1)^2 + C/(x+2)`

Leading to:

`2x -= A(x-1)(x+2) + B(x+2) + C(x-1)^2`

Where `A,B,C` are constants to be determined by substitution and/or comparing coefficients:

Put `x=1 => 2 = 3B => B =2/3`
Put `x=-2 => -4 = 9C => C =-4/9`

Equating Coefficients, we get:

`Coef(x^2) : 0 = A+C => A = 4/9`

So, we can write [A] as:

`int \ dy = int \ (4/9)/(x-1) + (2/3)/(x-1)^2 + (-4/9)/(x+2) \ dx`

`:. int \ dy = 4/9 \ int \ 1/(x-1) \ dx - 2/3 \ int \ (-1)/(x-1)^2 -4/9 \ int \ 1 /(x+2) \ dx`

And we can now readily integrate to get:

`y = 4/9 ln|x-1| - 2/3 \ 1/(x-1) -4/9 ln|x+2| + C`