Consider the differential equation \(\displaystyle\frac{dy}{dx} = \frac{y}{x} + 2\), \(x \gt 0\).
(A) It is a homogeneous differential equation.
(B) Its integrating factor is x.
(C) Its general solution is \(y = 2x\log x + Cx\), where C is an arbitrary constant.
(D) For the solution with \(y = 3\) when \(x = 1\), the value of y at \(x = e\) is 5e.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (C) and (D) only
The right side \(\frac{y}{x} + 2\) depends only on \(\frac{y}{x}\), so the equation is homogeneous. It is also linear:
\( \displaystyle \frac{dy}{dx} - \frac{1}{x}\,y = 2 \)
The integrating factor is
\( \displaystyle e^{-\log x} = \frac{1}{x} \)
So
\( \displaystyle \frac{y}{x} = \int \frac{2}{x}\,dx = 2\log x + C \)
\( y = 2x\log x + Cx \)
With \(y = 3\) at \(x = 1\): \(3 = 0 + C\), so \(C = 3\). At \(x = e\):
\( y = 2e + 3e = 5e \)
So the equation is homogeneous, its integrating factor is \(\frac{1}{x}\), its general solution is \(y = 2x\log x + Cx\), and the particular solution gives \(y = 5e\) at \(x = e\).
Checking the statements
(A) It is a homogeneous differential equation. — Correct. The right side is a function of \(\frac{y}{x}\) alone.
(B) Its integrating factor is x. — Incorrect. \(P = -\frac{1}{x}\), so the integrating factor is \(\frac{1}{x}\).
(C) Its general solution is \(y = 2x\log x + Cx\). — Correct. \(\frac{y}{x} = 2\log x + C\).
(D) For the solution with \(y = 3\) when \(x = 1\), the value of y at \(x = e\) is 5e. — Correct. \(C = 3\) and \(y(e) = 2e + 3e\).