The equation in the image is:

$x y' - y = y^3$

This is a first-order nonlinear differential equation because it involves the first derivative of $y$ with respect to $x$ (denoted as $y'$) and the cubic power of $y$.

Solution Method: Separation of Variables

We will attempt to solve this equation using the method of separation of variables.

The given equation is:

$x y' - y = y^3$

Step 1: Rewrite the equation in terms of $y'$:

$x y' = y + y^3$

Step 2: Divide both sides by $y + y^3$ to separate variables:

$\frac{y'}{y + y^3} = \frac{1}{x}$

Step 3: Simplify the left-hand side:

$\frac{1}{y + y^3} = \frac{1}{y(1 + y^2)}$

Step 4: Now integrate both sides:

$\int \frac{dy}{y(1 + y^2)} = \int \frac{dx}{x}$

The left side can be decomposed using partial fractions or standard techniques, and the right side is a straightforward natural logarithm.

Would you like me to continue the step-by-step integration and find the complete solution? Let me know!

Related Questions:

1. How does separation of variables work in solving differential equations?
2. Can nonlinear differential equations always be solved by separation of variables?
3. How do partial fractions help simplify the integration process?
4. What are other techniques for solving nonlinear first-order differential equations?
5. Can this equation be solved using an alternative method like an integrating factor?

Tip:

Always try simplifying differential equations by rewriting them in a form where you can apply standard methods, such as separation of variables or an integrating factor, to make the problem more manageable.