The general solution of the given differential equation \( \displaystyle\frac{dy}{dx}=\displaystyle\frac{3e^{2x}+3e^{4x}}{e^{x}+e^{-x}} \) (where C is an arbitrary constant)
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( y=e^{3x}+C \)
The right-hand side simplifies before integrating. Taking \( 3e^{2x} \) common in the numerator and \( e^{-x} \) common in the denominator:
\( \displaystyle \frac{3e^{2x}+3e^{4x}}{e^{x}+e^{-x}}=\frac{3e^{2x}(1+e^{2x})}{e^{-x}(e^{2x}+1)} \)
The brackets cancel:
\( \displaystyle \frac{dy}{dx}=\frac{3e^{2x}}{e^{-x}} \)
\( \displaystyle =3e^{3x} \)
\( \displaystyle y=\int 3e^{3x}\,dx \)
\( \displaystyle =3\cdot\frac{e^{3x}}{3}+C \)
\( \displaystyle =e^{3x}+C \)
Why the other options are incorrect
Option (1) \( y=3e^{3x}+C \) — Incorrect. This forgets to divide by 3 when integrating \( e^{3x} \); its derivative is \( 9e^{3x} \).
Option (2) \( y=\displaystyle\frac{1}{3}e^{3x}+C \) — Incorrect. This divides by 3 twice; its derivative is \( e^{3x} \), not \( 3e^{3x} \).
Option (3) \( y=9e^{3x}+C \) — Incorrect. This multiplies by 3 instead of dividing; its derivative is \( 27e^{3x} \).