Which of the following functions are solutions of the differential equation \(\displaystyle x\frac{dy}{dx} - y = x^2\)?
(A) \(y = x^2\)
(B) \(y = x^2 + 3x\)
(C) \(y = x^2 + 3\)
(D) \(y = 2x^2\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (B) only
A function is a solution if, after substituting y and dy/dx, the left side \(x\frac{dy}{dx} - y\) becomes exactly x2.
\(y = x^2\): \(x(2x) - x^2 = x^2\)
\(y = x^2 + 3x\): \(x(2x + 3) - x^2 - 3x = x^2\)
\(y = x^2 + 3\): \(x(2x) - x^2 - 3 = x^2 - 3\)
\(y = 2x^2\): \(x(4x) - 2x^2 = 2x^2\)
So the left side equals x2 for (A) and (B), but gives x2 − 3 for (C) and 2x2 for (D).
Checking the statements
(A) \(y = x^2\) — Correct. \(2x^2 - x^2 = x^2\).
(B) \(y = x^2 + 3x\) — Correct. \(2x^2 + 3x - x^2 - 3x = x^2\).
(C) \(y = x^2 + 3\) — Incorrect. The left side is \(x^2 - 3\), not x2.
(D) \(y = 2x^2\) — Incorrect. The left side is \(2x^2\), not x2.