Question

Given two functions, M(x, y) and N(x, y), suppose that (∂N/∂x − ∂M/∂y)/(M − N)
is a function of x + y. That is, let f(t) be a function such that

f(x + y) = (∂N/∂x − ∂M/∂y)/(M − N)
Assume that you can solve the differential equation
M dx + N dy = 0

by multiplying by an integrating factor μ that makes it exact and that it can also be
written as a function of x + y, μ = g(x + y) for some function g(t). Give a method
for finding this integrating factor μ, and use it to find the general solution to the
differential equation

(3 + y + xy)dx + (3 + x + xy)dy = 0.

Answer 1

Hello!!

Let be non- exact , which becomes exact after multiplying with integrating factor

. So, we have to an exact equation.

Thus,

[Note: The two important conditions are and (This condition will be needed later on)]

Now,

Also,

As, , we get

Integrating both sides,

or,

For given differential equation:

Here, and .

Now,

.

Now, the new equation is ,

Here, and .

Let be the solution for the differential equation.

So, and .

Here,

and .

Thus, the required function can be obtained by writing repeated terms only once.

Therefore, and , hence, the general solution is .

Hope this Helps!!

Enjoy!!