Let \(y = y(x)\) be the solution of the differential equation \(\displaystyle\frac{dy}{dx} = y\cos x\) satisfying \(y(0) = 3\).
(A) The general solution of the differential equation is \(y = Ce^{\sin x}\), C an arbitrary constant
(B) \(y\left(\displaystyle\frac{\pi}{2}\right) = 3e\)
(C) \(y(\pi) = 3e\)
(D) The greatest value of y(x) is 3e
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (D) only
The equation is variables separable:
\( \displaystyle \frac{dy}{y} = \cos x\,dx \)
\(\log|y| = \sin x + C_1\)
\(y = Ce^{\sin x}\)
Putting x = 0, y = 3 gives C = 3, since \(\sin 0 = 0\):
\(y = 3e^{\sin x}\)
So the general solution is \(y = Ce^{\sin x}\), the particular solution is \(y = 3e^{\sin x}\), y(π/2) = 3e, y(π) = 3, and the greatest value of y is 3e (when sin x = 1).
Checking the statements
(A) general solution \(y = Ce^{\sin x}\) — Correct. It comes from \(\log|y| = \sin x + C_1\).
(B) \(y\left(\frac{\pi}{2}\right) = 3e\) — Correct. \(\sin\frac{\pi}{2} = 1\), so \(y = 3e^{1}\).
(C) \(y(\pi) = 3e\) — Incorrect. \(\sin \pi = 0\), so \(y(\pi) = 3e^{0} = 3\).
(D) greatest value of y(x) is 3e — Correct. \(\sin x \le 1\), so \(3e^{\sin x} \le 3e\), reached at \(x = \frac{\pi}{2}\).