Introduction:
In this problem, we are given dy/dx and we need to find the value of x and y. The given expression can be written as (x^2 - 2y^2)/(2xy). To solve this, we need to use the concept of differential equations.

Solution:
To solve the given differential equation, we need to follow the following steps:

Step 1: Rewrite the given differential equation in the standard form of dy/dx + P(x)y = Q(x).

Step 2: Find the integrating factor (I.F.) of the given differential equation.

Step 3: Multiply both sides of the differential equation by the integrating factor.

Step 4: Integrate both sides of the differential equation with respect to x.

Step 5: Solve for y.

Step 1: Rewrite the given differential equation in the standard form of dy/dx + P(x)y = Q(x).

dy/dx = (x^2 - 2y^2)/(2xy)

Multiplying both sides by 2xy, we get:

2xy(dy/dx) = x^2 - 2y^2

Now, we can rearrange the terms to get:

2xy(dy/dx) + 2y^2 = x^2

Dividing both sides by 2y^2, we get:

(dy/dx)(x/y)^2 + 1 = (1/y^2)(x^2)

Now, we can rewrite the equation as:

dy/dx + (-x/y^2)y = x/y^2

Step 2: Find the integrating factor (I.F.) of the given differential equation.

The integrating factor (I.F.) for the given differential equation is:

I.F. = e^(∫(-x/y^2)dx) = e^(1/y)

Step 3: Multiply both sides of the differential equation by the integrating factor.

Multiplying both sides of the differential equation by the integrating factor, we get:

e^(1/y)dy/dx + e^(1/y)(-x/y^2)y = xe^(1/y)/y^2

Now, we can simplify the left-hand side using the product rule:

d/dx(e^(1/y)y) = xe^(1/y)/y^2

Step 4: Integrate both sides of the differential equation with respect to x.

Integrating both sides of the differential equation with respect to x, we get:

e^(1/y)y = ∫xe^(1/y)/y^2 dx + C

Step 5: Solve for y.

Now, we can solve for y by dividing both sides by e^(1/y):

y = e^(-1/y)∫xe^(1/y)/y^2 dx + Ce^(-1/y)

This is the general solution to the given differential equation.

Dy/dx=x²-2y²/2xy
2xydy/dx=x²-2y²
division with x²on both sides
2(y/x) dydx=1-2(y/x) ²
let y/x=v(differentiate on both sides)
dy/dx=v+x(dv/dx)
1-2v²=2v(v+xdv/dx)
simplified we get, 1-4v²=2vx(dv/dx)
Arrrange x and v terms on both sides
dx/x=2v/1-4v²(dv)
lnx=-1/4(ln[1-4v²]) +lnc
ln x+ln[ (1-4v²)] ¼=lnc
ln{(x) (1-4v²)¼}=lnc
x(1-4v²) =c
quadrating ob both sides
x⁴(1-4v²) =C , where(c⁴=C)
substitute v=y/x
x⁴(x²-4y²/x²) =C
x²(x²-4y²) =C
|x⁴-4x²y²-C=0|
is the required answer thank you