To solve the differential equation

$(4x + y)^2 \frac{dx}{dy} = 1,$

we start by separating the variables. Rearrange the equation to isolate $\frac{dx}{dy}$:

$\frac{dx}{dy} = \frac{1}{(4x + y)^2}.$

Now, we need to separate the variables $x$ and $y$. Rewrite the equation in the form suitable for separation:

$(4x + y)^2 \, dx = dy.$

Integrate both sides with respect to their respective variables:

$\int (4x + y)^2 \, dx = \int dy.$

To integrate the left-hand side, we'll use substitution. Let:

$u = 4x + y, \quad \text{so} \quad du = 4 \, dx \quad \text{or} \quad dx = \frac{du}{4}.$

Substitute into the integral:

$\int u^2 \frac{du}{4} = \int dy.$

Simplify and integrate:

$\frac{1}{4} \int u^2 \, du = y + C_1.$

$\frac{1}{4} \cdot \frac{u^3}{3} = y + C_1.$

Substitute back $u = 4x + y$:

$\frac{(4x + y)^3}{12} = y + C_1.$

Multiply by 12 to clear the fraction:

$(4x + y)^3 = 12y + C_2.$

where $C_2 = 12C_1$.

This is the general solution to the differential equation.

Would you like more details or have any questions? Here are some related questions that might interest you:

1. What is a differential equation and how do we solve them generally?
2. What other methods exist for solving differential equations?
3. How does separation of variables work for more complicated equations?
4. What are some practical applications of differential equations?
5. Can we solve this differential equation using another method, like an integrating factor?

Tip: When solving differential equations, always check if the variables can be separated easily before choosing a more complex method.