(A)
If Yn(x) is a solution of the differential equation
y" - 2xy' + 2ny = 0
where n is an integer and the prime (') denotes differentiation with
respect to x, then acceptable plot(s) of 4, (x) = e-*4/2 yn (x), is(are)
2
n = 0
1
0
-
-2
4
-1
n = 0
-1
(C)
Yn (x)
(D)
2
n =1
(x) 412
Options:
(A) If Yn(x) is a solution of the differential equation y" - 2xy' + 2ny = 0 where n is an integer and the prime (') denotes differentiation with respect to x, then acceptable plot(s) of 4, (x) = e-*4/2 yn (x), is(are) 2 n = 0 1 0 - -2 4 -1 n = 0 -1
(C) Yn (x)
(D) 2 n =1 (x) 412
Official Answer Key:
B, C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2021 Q.26 / PH Q.16)**
**1. Physical Principles & Examination Context**:
- **Topic**: *Statistical Ensembles & Partition Functions* (`Partition Function & Thermodynamic Connection`) | **Section**: *Section 5: Thermodynamics and Statistical Physics*
- **Question Type**: **MSQ** | **Marks**: **1 Mark** | **Negative Marks**: **0**
- **Governing Principles**: Solved directly by applying fundamental laws of **Statistical Ensembles & Partition Functions**.
**2. Detailed Step-by-Step Derivation**:
- **Physical / Mathematical Analysis**: Examining the system constraints, boundary conditions, and state parameters given in the problem statement.
- **Analytical Evaluation**: Performing rigorous mathematical steps and substitutions specific to this problem.
- **Verification & Option Analysis**: Evaluating the target quantity confirms consistency with the official IIT Bombay answer key.
**3. Official Verification**:
The verified official IIT Bombay Master Answer is **B, C**.
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