GATE 2022 PH – Question 19

Correct answer: (B) $c=-2a\ne0,b=0$

Explanation

Substitute $y=a+bx+cx^2$ into $y^{\prime\prime}-2xy^\prime+4y=0$.

• $y^\prime=b+2cx$ and $y^{\prime\prime}=2c$.
• $-2xy^\prime=-2bx-4cx^2$.
• $4y=4a+4bx+4cx^2$.

**Sum:**
$$2c+4a+(-2b+4b)x+(-4c+4c)x^2=(2c+4a)+2bx=0 .$$

For this to vanish for all $x$:
- the coefficient of $x$: $2b=0\Rightarrow b=0$,
- the constant: $2c+4a=0\Rightarrow c=-2a$.

So $c=-2a\neq0$ and $b=0$ (option B).