If the solution curve of the differential equation \(\displaystyle\frac{dy}{dx} + 2y = e^{3x}\), passes through the point \(\left(0, \displaystyle\frac{6}{5}\right)\), then the value of \(y(\log_e 2)\) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{37}{20}\)
This is a linear differential equation with P = 2 and Q = \(e^{3x}\).
\(\text{I.F.} = e^{\int 2\,dx} = e^{2x}\)
The solution is
\( \displaystyle y\,e^{2x} = \int e^{3x}\cdot e^{2x}\,dx \)
\( \displaystyle = \frac{e^{5x}}{5} + C \)
At \(\left(0, \frac{6}{5}\right)\):
\( \displaystyle \frac{6}{5} = \frac{1}{5} + C,\quad C = 1 \)
Dividing by \(e^{2x}\):
\( \displaystyle y = \frac{e^{3x}}{5} + e^{-2x} \)
At \(x = \log_e 2\), \(e^{3x} = 8\) and \(e^{-2x} = \frac{1}{4}\):
\( \displaystyle y(\log_e 2) = \frac{8}{5} + \frac{1}{4} \)
\( \displaystyle = \frac{32 + 5}{20} = \frac{37}{20} \)