The general solution of the differential equation \( \displaystyle\frac{dy}{dx}=\displaystyle\frac{3e^{2x}+3e^{4x}}{e^{x}+e^{-x}} \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( y=e^{3x}+\mathrm{C} \), where \( c \) is arbitrary constant
The right-hand side simplifies once \( 3e^{2x} \) is taken common in the numerator and \( e^{x}+e^{-x} \) is written over the denominator \( e^{x} \):
\( \displaystyle \frac{3e^{2x}+3e^{4x}}{e^{x}+e^{-x}}=\frac{3e^{2x}\left(1+e^{2x}\right)}{\frac{e^{2x}+1}{e^{x}}} \)
\( \displaystyle =3e^{2x}\cdot e^{x} \)
\( \displaystyle =3e^{3x} \)
So the equation is in variables separable form:
\( \displaystyle \frac{dy}{dx}=3e^{3x} \)
\( \displaystyle y=\int 3e^{3x}\,dx \)
\( \displaystyle =3\cdot\frac{e^{3x}}{3}+\mathrm{C} \)
\( \displaystyle =e^{3x}+\mathrm{C} \)
Why the other options are incorrect
Option (2) \( y=\displaystyle\frac{e^{3x}}{3}+\mathrm{C} \), where \( c \) is arbitrary constant — Incorrect. This drops the 3 of the numerator; the factor 3 in \( 3e^{3x} \) cancels the \( \frac{1}{3} \) from integrating \( e^{3x} \).
Option (3) \( y=3e^{ex}+\mathrm{C} \), where \( c \) is arbitrary constant — Incorrect. Differentiating \( 3e^{ex} \) gives \( 3e\cdot e^{ex} \), which is not \( 3e^{3x} \).
Option (4) \( y=\displaystyle\frac{e^{2x}}{2}+\mathrm{C} \), where \( c \) is arbitrary constant — Incorrect. Differentiating \( \frac{e^{2x}}{2} \) gives \( e^{2x} \); the factor \( e^{x} \) from the denominator has been ignored.