Which of the following differential equations has \( y=x \) as one of the particular solution?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{d^2y}{dx^2}-x^2\displaystyle\frac{dy}{dx}+xy=0 \)
For y = x,
\( \displaystyle \frac{dy}{dx}=1 \)
\( \displaystyle \frac{d^2y}{dx^2}=0 \)
A particular solution must satisfy the equation for all x. Substituting in each equation:
Option (1) \( \displaystyle\frac{d^2y}{dx^2}-x\displaystyle\frac{dy}{dx}+xy=x \) — Incorrect. The left side is \( 0-x+x^2 \), which equals x only when \( x^2=2x \).
Option (2) \( \displaystyle\frac{d^2y}{dx^2}+x\displaystyle\frac{dy}{dx}+xy=x \) — Incorrect. The left side is \( 0+x+x^2 \), which equals x only when \( x^2=0 \).
Option (3) \( \displaystyle\frac{d^2y}{dx^2}-x^2\displaystyle\frac{dy}{dx}+xy=0 \) — Correct. The left side is \( 0-x^2+x^2=0 \) for every x.
Option (4) \( \displaystyle\frac{d^2y}{dx^2}+x\displaystyle\frac{dy}{dx}+xy=0 \) — Incorrect. The left side is \( x+x^2 \), which is not 0 for all x.