Solving the Differential Equation (dy/dx 1/(e^y - x))

The given differential equation is:

To solve this, we can use the method of separation of variables. The first step is to rearrange the equation to separate the variables y and x.

Separate Variables

We can rewrite the equation as:

dx ey - x dy

Integrate Both Sides

Now, we integrate both sides. The left side requires integration with respect to y, and the right side requires integration with respect to x:

int ey - x dy int dx

Perform the Integrals

- The left integral becomes:

int ey dy - int x dy ey - xy C1

- The right integral is:

int dx x C2

We can combine the constants C1 and C2 into a single constant C:

ey - xy x C

Rearrange

To express the solution in a more standard form, we have:

ey xy xC

This equation represents the implicit solution to the given differential equation. Depending on the context, you might solve explicitly for y if needed, but this form is generally sufficient for most applications.

Alternative Solution: Solving for x

We have the first method, but if we want to solve for x, we can follow these steps:

Multiply both sides by integrating factor e∫1 dy ey:

eydx/dy - xey e2y

y) e2y}

Integrating Both Sides

int d(xey) int e2y dy

ixey 1/2 e2y A

x 1/2 ey Aey

(Where A is the integrating constant.)

Alternative Solution: Solving for y

If we want to solve for y, we can multiply both sides by ey:

dx/dyx ey x

d(xey)/dy e2y}

int d(xey) int e2y dy

xey 1/2 e2y C

x 1/2 ey Cey

(Where C is the integrating constant.)

For further simplification, we can rearrange to solve for y:

2xey e2y C

e2y - 2xey C 0

ey x pm; sqrt(x2 - C)

y ln(x pm; sqrt(x2 - C))

Thus, the differential equation has been solved and its solution provided formally.