Question

Which of the following equations is NOT a Linear Differential Equation?

• A. \((1 + x^2) \, dy + 2xy \, dx = \cot x \, dx\)
• B. \(y + \frac{d}{dx}(xy) = x(\sin x + \log x)\)
• C. \(x(1 + y^2) \, dx - y(1 + x^2) \, dy = 0\)
• D. \(y \, dx - (x + 3y^2) \, dy = 0\)

Hint:

To quickly check for linearity, look for powers of the dependent variable or its derivatives. If you spot terms like \(y^2\), \(\sin(y)\), or \(e^y\), and the equation cannot be rearranged to treat \(x\) as a linear dependent variable, it is non-linear.

Answer 1

The Correct Option is C

To determine which of the given equations is not a linear differential equation, we must first understand what constitutes a linear differential equation. A linear differential equation is one in which the dependent variable and its derivatives appear to the power of one (i.e., are linear) and are not multiplied together.

Let's analyze each option:

1. (1 + x^2) \, dy + 2xy \, dx = \cot x \, dx

   This is a linear differential equation since the terms involving the dependent variable y and its derivative dy are linear, and they are not multiplied together.

2. y + \frac{d}{dx}(xy) = x(\sin x + \log x)

   This equation can be rewritten to isolate the derivative:

   \frac{d}{dx}(xy) = x\sin x + x\log x - y

   It is linear in terms of y because y appears only once without being multiplied by another y term.

3. x(1 + y^2) \, dx - y(1 + x^2) \, dy = 0

   This can be rewritten as:

   x(1 + y^2) \, dx = y(1 + x^2) \, dy

   Upon closer inspection, this equation is nonlinear because it contains the term y^2. This makes it a nonlinear differential equation, as the dependent variable y appears with a power greater than one.

4. y \, dx - (x + 3y^2) \, dy = 0

   Although this equation includes the term 3y^2, the structure of the equation suggests it is intended to be part of a differential form rather than defining linearity. However, technically, the 3y^2 term indicates nonlinearity because of the y^2 term.

From this analysis, the equation x(1 + y^2) \, dx - y(1 + x^2) \, dy = 0 is the correct choice as it clearly contains nonlinear terms involving y^2.