Step 1 (Formula / Governing Law & Given Data): [0.25 Mark]

A differential equation is defined as a Linear Differential Equation if the dependent variable and its derivatives appear only in the first degree and are not multiplied together. The standard forms are:

• Form 1: $\frac{dy}{dx} + Py = Q$, where $P$ and $Q$ are functions of $x$ only.
• Form 2: $\frac{dx}{dy} + P_1x = Q_1$, where $P_1$ and $Q_1$ are functions of $y$ only.

Step 2 (Step-by-Step Mathematical Working): [0.5 Mark]

We analyze each option to check for linearity:

• (A) $(1 + x^2) dy + 2xy dx = \cot x dx \implies \frac{dy}{dx} + \frac{2x}{1+x^2}y = \frac{\cot x}{1+x^2}$. This is linear in $y$.
• (B) $y + x\frac{dy}{dx} + y = x(\sin x + \log x) \implies x\frac{dy}{dx} + 2y = x(\sin x + \log x)$. This is linear in $y$.
• (C) $x(1 + y^2) dx = y(1 + x^2) dy \implies \frac{dx}{dy} = \frac{y(1+x^2)}{x(1+y^2)}$. Here, the variable $y$ is squared ($y^2$), and the equation cannot be rearranged into the form $\frac{dx}{dy} + P_1x = Q_1$ because of the $y^2$ term multiplying the differential.
• (D) $y dx = (x + 3y^2) dy \implies \frac{dx}{dy} = \frac{x}{y} + 3y \implies \frac{dx}{dy} - \frac{1}{y}x = 3y$. This is linear in $x$.

Step 3 (Final Concluded Answer): [0.25 Mark]

Since option (C) contains a $y^2$ term that prevents it from being expressed in the standard linear form, it is not a Linear Differential Equation.

Correct Option: (C)

💡 EL10 Concept Insight: Think of a "Linear" equation like a straight line—it doesn't allow variables to be squared, cubed, or multiplied by each other (like $y^2$ or $xy$). Because option (C) has a $y^2$ term, it creates a "curve" in the math, making it non-linear!