GATE 2026 PH – Question 50

Mathematical Physics · linear differential equations: simple applications of first and second order linear differential equations and solutions · 2 marks · Multiple select

Which of the following operators is/are self-adjoint?

$x^2\frac{d^2}{dx^2}+3x\frac d{dx}+x^2$
$(1-x^2)\frac{d^2}{dx^2}-2x\frac d{dx}+3x$
$(3x-4x^3)\frac{d^2}{dx^2}+(3-12x^2)\frac d{dx}+12$
$x\frac{d^2}{dx^2}+x^2\frac d{dx}+5x/3$

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Correct answer: (B) $(1-x^2)\frac{d^2}{dx^2}-2x\frac d{dx}+3x$; (C) $(3x-4x^3)\frac{d^2}{dx^2}+(3-12x^2)\frac d{dx}+12$

Explanation

A real second-order differential operator
$$\hat L=a(x)\frac{d^2}{dx^2}+b(x)\frac{d}{dx}+c(x)$$
is **formally self-adjoint** (with a unit weight function) when $b(x)=a^\prime(x)$, because then it can be written in the Sturm-Liouville form $\hat L=\dfrac{d}{dx}\left(a\dfrac{d}{dx}\right)+c$.

Check each operator:
- **A.** $a=x^2$, $a^\prime=2x$, but $b=3x$. ✗
- **B.** $a=1-x^2$, $a^\prime=-2x$, and $b=-2x$. ✓
- **C.** $a=3x-4x^3$, $a^\prime=3-12x^2$, and $b=3-12x^2$. ✓
- **D.** $a=x$, $a^\prime=1$, but $b=x^2$. ✗

Answer **B and C**. (A and D could be made self-adjoint only with a different weight function.)

Official GATE 2026 answer key: https://gate2026.iitg.ac.in/doc/download/2026/Keys/PH_Keys.pdf