Consider the differential equation $\frac{dy}{dx}=xy+x$. Which of the following best describes this equation?

• A. It is neither linear nor separable.
• B. It is both linear and separable.
• C. It is separable but not linear.
• D. It is linear but not separable.

Answer: (D)

Verified step by step guidance

1. Step 1: Understand the definition of a linear differential equation. A first-order linear differential equation has the form: $dy/dx\; +\; P(x)y\; =\; Q(x)$, where P(x) and Q(x) are functions of x.
2. Step 2: Rewrite the given equation $dy/dx\; =\; xy\; +\; x$. Observe that it can be rearranged as $dy/dx\; -\; xy\; =\; x$, which matches the form of a linear differential equation with $P(x)\; =\; -x$ and $Q(x)\; =\; x$.
3. Step 3: Determine if the equation is separable. A separable differential equation can be written in the form $dy/dx\; =\; g(x)h(y)$, where the variables x and y can be separated into two distinct functions. In this case, $dy/dx\; =\; xy\; +\; x$ cannot be separated into such a form because the term $x$ is added to $xy$, making separation impossible.
4. Step 4: Conclude that the equation is linear because it fits the form of a linear differential equation, but it is not separable because the variables cannot be separated into distinct functions of x and y.
5. Step 5: Verify the classification of the equation. Based on the analysis, the correct description is: 'It is linear but not separable.'