The image contains two differential equations:

1. $(x^2 - yx^2) \, dy + (y^2 + xy^2) \, dx = 0$
2. $x^2 \, dy - (2xy + 3y) \, dx = 0$

Let’s analyze and solve these equations step by step.

1. Solving the First Equation:

$(x^2 - yx^2) \, dy + (y^2 + xy^2) \, dx = 0$

Simplification:

• Factor terms where possible: $x^2(1 - y) \, dy + y^2(1 + x) \, dx = 0$

Identify as a homogeneous differential equation:

This equation can be analyzed for homogeneity since all terms involve $x$ and $y$. Rewrite it in the form: $\frac{dy}{dx} = -\frac{y^2(1 + x)}{x^2(1 - y)}.$

At this point, we can solve the equation using substitution methods for homogeneous functions.

2. Solving the Second Equation:

$x^2 \, dy - (2xy + 3y) \, dx = 0$

Rearrange into standard form:

$\frac{dy}{dx} = \frac{2xy + 3y}{x^2}.$

Simplify:

Factor $y$ from the numerator: $\frac{dy}{dx} = \frac{y(2x + 3)}{x^2}.$

Use separation of variables:

$\frac{dy}{y} = \frac{(2x + 3)}{x^2} \, dx.$

Separate and integrate both sides: $\int \frac{1}{y} \, dy = \int \frac{2x}{x^2} \, dx + \int \frac{3}{x^2} \, dx.$

The left-hand side integrates to $\ln |y|$. The right-hand side simplifies as follows: $\int \frac{2x}{x^2} \, dx = 2 \int \frac{1}{x} \, dx = 2 \ln |x|,$ $\int \frac{3}{x^2} \, dx = -\frac{3}{x}.$

Combining results: $\ln |y| = 2 \ln |x| - \frac{3}{x} + C,$ where $C$ is the constant of integration.

Final Solution for the Second Equation:

Exponentiate both sides to isolate $y$: $y = Cx^2 e^{-3/x},$ where $C$ is an arbitrary constant.

Would you like me to solve the first equation in more detail or explain a particular step? Let me know!

Related Questions:

1. How do you solve homogeneous differential equations step by step?
2. What is the method of separation of variables in differential equations?
3. How do you check if a first-order equation is exact?
4. What are the properties of integrating factors in differential equations?
5. How do you solve non-linear first-order differential equations?

Tip: For equations involving $x$ and $y$, always simplify terms first to check if they are separable, homogeneous, or exact.