## Solving the Differential Equation: (xy^2 + 2x^2y^3)dx + (x^2y - x^3y^2)dy = 0

This differential equation is an example of a **non-exact** differential equation. This means it can't be directly integrated to find a solution. To solve it, we'll employ a strategy called **finding an integrating factor**.

**1. Checking for Exactness:**

A differential equation of the form M(x,y)dx + N(x,y)dy = 0 is exact if ∂M/∂y = ∂N/∂x.

In our case:

- M(x,y) = xy^2 + 2x^2y^3
- N(x,y) = x^2y - x^3y^2

Calculating the partial derivatives:

- ∂M/∂y = 2xy + 6x^2y^2
- ∂N/∂x = 2xy - 3x^2y^2

Since ∂M/∂y ≠ ∂N/∂x, the equation is not exact.

**2. Finding an Integrating Factor:**

We'll look for an integrating factor μ(x,y) that makes the equation exact. There are two common approaches:

**a) μ(x) depending only on x:**

If (∂N/∂x - ∂M/∂y)/M is a function of x only, then μ(x) = exp(∫(∂N/∂x - ∂M/∂y)/M dx) is an integrating factor.

In our case, (∂N/∂x - ∂M/∂y)/M = (-9x^2y^2)/(xy^2 + 2x^2y^3) = -9x/y(1 + 2xy) is not a function of x only. So this approach won't work.

**b) μ(y) depending only on y:**

If (∂M/∂y - ∂N/∂x)/N is a function of y only, then μ(y) = exp(∫(∂M/∂y - ∂N/∂x)/N dy) is an integrating factor.

Let's check: (∂M/∂y - ∂N/∂x)/N = (9x^2y^2)/(x^2y - x^3y^2) = 9/y(1 - xy)

This is a function of y only. Therefore, we can find an integrating factor μ(y):

μ(y) = exp(∫9/y(1 - xy) dy) = exp(9ln(y) - 9∫y/(1-xy) dy)

We can solve the integral using u-substitution (u = 1-xy) and obtain:

μ(y) = y^9/(1 - xy)^9

**3. Multiplying by the Integrating Factor:**

Multiply both sides of the original equation by μ(y):

(y^9/(1 - xy)^9)(xy^2 + 2x^2y^3)dx + (y^9/(1 - xy)^9)(x^2y - x^3y^2)dy = 0

**4. Checking for Exactness (Again):**

Now the equation is exact since:

- ∂(Mμ)/∂y = ∂(Nμ)/∂x = 9x^2y^8/(1-xy)^10

**5. Solving the Exact Equation:**

We can now find a solution by integrating:

∫(y^9/(1 - xy)^9)(xy^2 + 2x^2y^3)dx = F(x,y) = C

The integral on the left can be solved using u-substitution (u = 1-xy) and partial fraction decomposition. After integrating and simplifying, we'll find:

F(x,y) = -x^2y^9/(1 - xy)^8 + C

**Conclusion:**

The general solution to the differential equation (xy^2 + 2x^2y^3)dx + (x^2y - x^3y^2)dy = 0 is given by:

**-x^2y^9/(1 - xy)^8 = C**

Where C is an arbitrary constant.