Solve the given differential equation by separation of variables. dy dx = xy + 5x - y - 5 xy - 2x + 6y - 12

solution:

Consider the differential equation DE

Dy/dx = xy + 5x - y - 5 / xy - 2x + 6y - 12

Write the DE as the follows.

Dy/dx = x (y+5) - 1 (y+5) / x (y-2) + 6 (y-2)

Dy/dx = (x-1) (y+5) / (x+6) (y-2)

Separate the variables.

y-2/y+5 dy = x-1/x+6 dx

integrate on the both sides,

∫y-2/y+5 dy = ∫x-1/x+6 dx

7in (x+6) - 7in (y+5) = x-y+c

In (x+6) ∧7 - in (y+5) ∧7 = x-y+c, using bIna = Inab

In [ (x+6) ∧7 / (y+5) ∧7] = x-y+c, using Ina - Inb = In (b/a)

eIn[ (x+6) ∧7 / (y+5) ∧7] = ex-y+c, taking exponents on both sides

(x+6) ∧7 / (y+5) ∧7 = ec. ex-y, use eInx = x

(x+6) ∧7 / (y+5) ∧7 = c1ex-y, take ec = c1

Hence, the solution of the DE is (x+6) ∧7 / (y+5) ∧7 = c1ex-y