Question

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

Answer 1

The differential equation provided seems to contain a typo and cannot be solved as is. Once the correct form of the equation is provided, the solution involves separating the variables, integrating both sides, and finding the relation between x and y.

Step-by-step explanation:

To solve the given differential equation by separation of variables, we first need to rewrite it in a form that clearly separates the variables x and y. However, the equation provided seems to have a typo and should likely be in the form dy/dx = f(x, y) where f(x, y) is some function of both x and y. If we assume that the given equation is of the form (xy + 5x - y - 5)/(xy - 2x + 6y - 12), the separation of variables technique would involve rearranging the equation so that all terms involving y are on one side and all terms involving x are on the other side.

Unfortunately, without the correct form of the equation, a specific solution cannot be provided. If the student can provide the correct differential equation, the following steps will be taken to solve it:

• Rearrange the equation to separate x and y variables.
• Integrate both sides with respect to their respective variables.
• Apply initial conditions if any are given or solve for the arbitrary constants.
• Arrive at the solution y in terms of x or a relation between x and y.

Answer 2

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