Question

Consider the partial differential equation: \[ \frac{\partial^2 u}{\partial x^2} = \frac{1}{k} \frac{\partial u}{\partial t} + \sin x, \quad k>0 \] Amongst the following, the correct statement(s) for the above equation is/are:

• A. It is homogeneous
• B. It is linear
• C. It is of degree 1
• D. It is of order 2

Hint:

For partial differential equations, remember: - Homogeneity refers to the presence or absence of non-homogeneous terms (e.g., constants or functions independent of \( u \)). - Linearity refers to the dependent variable and its derivatives appearing to the first power and not multiplied together. - The degree is the highest power of the highest order derivative. - The order is the highest derivative in the equation.

Answer 1

The Correct Option is B, C, D

Step 1: Identifying Homogeneity. The equation has a non-homogeneous term \( \sin x \) on the right-hand side. A homogeneous partial differential equation should not have terms that do not depend on the dependent variable \( u \) or its derivatives. Since the equation includes \( \sin x \), which is independent of \( u \), it is not homogeneous. 
Step 2: Identifying Linearity. The equation is linear if the dependent variable \( u \) and its derivatives appear to the first power and are not multiplied together. Here, both \( \frac{\partial^2 u}{\partial x^2} \) and \( \frac{\partial u}{\partial t} \) appear linearly (i.e., they are not products or powers of \( u \) and its derivatives). Thus, the equation is linear. 
Step 3: Identifying Degree. The degree of a partial differential equation refers to the highest power of the highest order derivative of the dependent variable in the equation. The highest order derivative in this case is \( \frac{\partial^2 u}{\partial x^2} \), and it appears to the first power. Therefore, the equation is of degree 1. 
Step 4: Identifying Order. The order of a partial differential equation is determined by the highest order derivative. The highest order derivative in this equation is \( \frac{\partial^2 u}{\partial x^2} \), which is of order 2. 
Therefore, the equation is of order 2.