Partial Differential equation problem

[eljose]

let be the function U=U(x,y) satisfying:
$$U_{x}=(a/\epsilon)^{y}y$$ and $$U_{y}=(a/\epsilon)^{x}x$$ (1)
where we have introduced the notation $$U_{i}=dU/di$$ i=x,y then from expression (1) we could construct the differential equation:
$$xU_{x}-yU_{y}=0$$ (2)
from (2) we could construct the solution to obtain U, my problem is how to obtain U so its derivatives respect to x and y give the result in (1)

 

[Tide]

How does (2) follow from (1)?

 

[tacman]

To obtain the solution for U, we can use the method of separation of variables. Let us assume that U can be written as a product of two functions, one dependent on x and the other on y, i.e. U(x,y) = X(x)Y(y).

Substituting this into (1), we get:
X'(x)Y(y) = (a/ε)^y y
and
X(x)Y'(y) = (a/ε)^x x

Dividing the first equation by X(x) and the second equation by Y(y), we get:
X'(x)/X(x) = (a/ε)^y y
and
Y'(y)/Y(y) = (a/ε)^x x

These are two ordinary differential equations, one in terms of x and the other in terms of y. We can solve these separately to obtain X(x) and Y(y). Let's solve the first equation in terms of x:
X'(x)/X(x) = (a/ε)^y y

Integrating both sides with respect to x, we get:
ln(X(x)) = (a/ε)^y yx + C1
where C1 is the constant of integration.

Taking the exponential of both sides, we get:
X(x) = e^[(a/ε)^y yx + C1]
= e^(C1) * e^[(a/ε)^y yx]

Let's define a new constant, C2 = e^(C1). Then we have:
X(x) = C2 * e^[(a/ε)^y yx]

We can do a similar process for the second equation in terms of y and obtain:
Y(y) = C3 * e^[(a/ε)^x xy]

Now, we can combine these two solutions to obtain the solution for U:
U(x,y) = X(x)Y(y)
= C2 * e^[(a/ε)^y yx] * C3 * e^[(a/ε)^x xy]
= C4 * e^[(a/ε)^y yx + (a/ε)^x xy]

where C4 = C2 * C3 is a new constant.

This is the general solution for U. To obtain the specific solution that satisfies (1), we need to find the value of C4. Substituting U(x,y)