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