# \ #
# "We will solve the differential equation given, and then look to" #
# "determine any unknown constants that may be in the result." #
# "We are given the following differential equation:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad dy/dx \ = \ x ( 2 a - 4/x ). #
# \qquad \qquad \qquad :. \qquad \qquad \ \ dy/dx \ = \ 2 a x - 4. #
# \quad \ :. \qquad \qquad \ \ \int \ dy/dx \ dx\ = \ int \ ( 2 a x - 4 ) dx. #
# \qquad \qquad \quad :. \qquad \qquad \ \ \int \ dy \ = \ int \ ( 2 a x - 4 ) dx. #
# \qquad \qquad \qquad :. \qquad \qquad \ \ y \ = \ a x^2 - 4 x + C. \qquad \qquad \qquad \qquad \qquad \qquad \qquad (1) #
# "We are given the solution curve passes through these points:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad (-1, 3), \qquad (2, -6). \qquad #
# "So we can substitute each of these points into (1):" #
# \qquad \qquad \qquad (-1, 3) \ rArr \qquad \ \ 3 \ = \ a (-1)^2 - 4 (-1) + C. #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad :. \qquad \qquad 3 \ = \ a + 4 + C. #
# \qquad \qquad \qquad (2, -6) \ rArr -6 \ = \ a (2)^2 - 4 (2) + C. #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad :. \ \ \ -6 \ = \ 4 a - 8 + C. #
# "Thus, we have:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \quad \qquad 3 \ = \ a + 4 + C. #
# \qquad \qquad \qquad \qquad \qquad \qquad \ -6 \ = \ 4 a - 8 + C. #
# "So:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \quad \qquad a + C \ = \ -1. #
# \qquad \qquad \qquad \qquad \qquad \qquad \quad \ \ 4 a + C \ = \ 2. #
# "The solutions are very easily seen to be:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \quad \qquad a \ = \ 1. #
# \qquad \qquad \qquad \qquad \qquad \qquad \quad \ \ \ C \ = \ -2. #
# "Substituting these values into (1) above, gives us the equation" ## "of the curve:" #
# \qquad \qquad \qquad :. \qquad \qquad \ \ y \ = \ (1) x^2 - 4 x + (-2). #
# \qquad \qquad \qquad :. \qquad \qquad \ \ y \ = \ x^2 - 4 x - 2. #
# "This is the equation of our desired curve." \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ square #
