The equation you've provided is:

$x y^2 + 3 y^2 - x^2 y' = 0$

Let's attempt to solve this differential equation using the method of separation of variables.

Step 1: Rewriting the equation

We first rewrite the equation by isolating the derivative $y'$ on one side. Starting from:

$x y^2 + 3 y^2 - x^2 y' = 0$

We can group terms involving $y'$ and move everything else to the other side:

$- x^2 y' = - x y^2 - 3 y^2$

Which simplifies to:

$x^2 y' = x y^2 + 3 y^2$

Step 2: Factor out common terms

We can factor out $y^2$ from the right-hand side:

$x^2 y' = y^2 (x + 3)$

Step 3: Separation of variables

Now we separate the variables $y$ and $x$. Dividing both sides by $y^2$ and multiplying both sides by $dx$, we get:

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

Step 4: Integration

We now integrate both sides:

• The left-hand side:

$\int \frac{dy}{y^2} = -\frac{1}{y}$

• The right-hand side:

$\int \frac{x + 3}{x^2} dx = \int \left( \frac{1}{x} + \frac{3}{x^2} \right) dx = \ln|x| - \frac{3}{x}$

Thus, after integration, we have:

$-\frac{1}{y} = \ln|x| - \frac{3}{x} + C$

Where $C$ is the constant of integration.

Step 5: Solve for $y$

Finally, solve for $y$ by inverting the equation:

$y = -\frac{1}{\ln|x| - \frac{3}{x} + C}$

This gives the general solution to the differential equation.

Would you like more details or clarifications?

Related Questions:

1. How does the method of separation of variables work for different types of differential equations?
2. Can this equation be solved using another method, like an integrating factor?
3. How would the solution change if the constant $C$ were to be zero?
4. What is the significance of the natural logarithm in the solution?
5. How do boundary conditions affect the solution of differential equations?

Tip:

When solving differential equations, always verify whether initial or boundary conditions are given—they can determine the specific solution from the general form.