To determine whether the given differential equation is exact, we need to check if it satisfies the condition for exactness. The general form of a first-order differential equation is:

$M(x, y) \, dx + N(x, y) \, dy = 0$

where $M(x, y) = x - y^7 + y^2 \sin(x)$ and $N(x, y) = 7xy^6 + 2y \cos(x)$.

The condition for exactness is:

$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$

Step 1: Compute $\frac{\partial M}{\partial y}$

First, calculate the partial derivative of $M(x, y)$ with respect to $y$:

$M(x, y) = x - y^7 + y^2 \sin(x)$

Taking the derivative with respect to $y$:

$\frac{\partial M}{\partial y} = -7y^6 + 2y \sin(x)$

Step 2: Compute $\frac{\partial N}{\partial x}$

Next, calculate the partial derivative of $N(x, y)$ with respect to $x$:

$N(x, y) = 7xy^6 + 2y \cos(x)$

Taking the derivative with respect to $x$:

$\frac{\partial N}{\partial x} = 7y^6 - 2y \sin(x)$

Step 3: Check if the equation is exact

Now, we compare $\frac{\partial M}{\partial y}$ with $\frac{\partial N}{\partial x}$:

$\frac{\partial M}{\partial y} = -7y^6 + 2y \sin(x)$ $\frac{\partial N}{\partial x} = 7y^6 - 2y \sin(x)$

Since $\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}$, the differential equation is not exact.

Thus, the answer is:

$\boxed{\text{NOT}}$

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If you'd like to explore this topic further, here are some questions you could consider:

1. What does it mean for a differential equation to be exact?
2. How can we make a non-exact differential equation exact?
3. What is the method of solving exact equations once the condition is satisfied?
4. How can you check if an equation is exact using an integrating factor?
5. What are some examples of exact differential equations?

Tip: Always ensure to check for the exactness condition before proceeding to solve the equation directly.