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
