Question

Which of the following is not a non-linear partial differential equation, where \(p = \frac{\partial z}{\partial x} \quad \text{and} \quad q = \frac{\partial z}{\partial y}?\)

• A. \( p + q = pq \)
• B. \( x^2p^2 + y^2q^2 = z^2 \)
• C. \( (x + y)\left(\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y}\right)^2 + (x - y)\left(\frac{\partial z}{\partial x} - \frac{\partial z}{\partial y}\right)^2 = 1 \)
• D. \( \frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial x \partial y} - 6 \frac{\partial^2 z}{\partial y^2} = x^2 \sin(x + y) \)

Hint:

In non-linear partial differential equations the derivatives of the function are raised to powers or multiplied together

Answer 1

The Correct Option is D

Option 4's equation is a linear partial differential equation, containing only second-order derivatives with respect to x and y and no terms where p and q are multiplied. The other equations are non-linear due to the inclusion of terms such as pq or powers of p and q.