Question

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : If \(u=x^n f\left(\dfrac{y}{x}\right)\), then \[ x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=nu \] Reason (R) : Given function \(u\) is homogeneous of degree \(n\) in \(x\) and \(y\). In the light of the above statements, choose the most appropriate answer from the options given below :

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

Whenever a function is of the form: \[ u=x^n f\left(\frac{y}{x}\right) \] it is automatically homogeneous of degree \(n\). Then Euler’s theorem can be directly applied: \[ x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=nu \]

Answer 1

The Correct Option is A

Concept: This problem is based on Euler’s Theorem on Homogeneous Functions. If a function \(u(x,y)\) is homogeneous of degree \(n\), then: \[ x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=nu \] A function is called homogeneous of degree \(n\) if: \[ u(tx,ty)=t^n u(x,y) \] Functions of the form: \[ u=x^n f\left(\frac{y}{x}\right) \] are always homogeneous functions of degree \(n\).

Step 1: Understanding the given function. We are given: \[ u=x^n f\left(\frac{y}{x}\right) \] Notice carefully that: \[ \frac{y}{x} \] is dimensionless with respect to scaling. If we replace: \[ x\to tx,\qquad y\to ty \] then: \[ \frac{ty}{tx}=\frac{y}{x} \] Hence: \[ u(tx,ty) = (tx)^n f\left(\frac{ty}{tx}\right) \] \[ =t^n x^n f\left(\frac{y}{x}\right) \] \[ =t^n u(x,y) \] Therefore \(u\) is homogeneous of degree \(n\).

Step 2: Applying Euler’s theorem. Since \(u\) is homogeneous of degree \(n\), Euler’s theorem gives: \[ x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} = nu \] This exactly matches Assertion (A).

Step 3: Checking Assertion (A). Assertion (A) states: \[ x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} = nu \] We have proved this using Euler’s theorem. Hence Assertion (A) is correct.

Step 4: Checking Reason (R). Reason (R) states: \[ u \text{ is homogeneous of degree } n \] We already established: \[ u(tx,ty)=t^n u(x,y) \] Hence Reason (R) is also correct.

Step 5: Checking whether Reason explains Assertion. Euler’s theorem applies exactly because the function is homogeneous of degree \(n\). Thus the reason directly explains the assertion. Therefore: \[ \boxed{ \text{Both (A) and (R) are correct and (R) is the correct explanation of (A)} } \] Hence the correct option is: \[ \boxed{(1)} \]