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

The equation in option 4 is a linear partial differential equation as it only involves second-order derivatives with respect to x and y and does not contain terms where p and q are multiplied The other equations are non-linear due to the presence of terms like pq or powers of p and q