If \(y = y(x)\) is the solution of the differential equation \(\displaystyle\frac{dy}{dx} = ye^{x}\) satisfying \(y(0) = 1\), then \(y(\log_e 2)\) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(e\)
The equation is variables separable (y stays positive, since y(0) = 1):
\( \displaystyle \frac{dy}{y} = e^{x}\,dx \)
Integrating both sides:
\(\log_e y = e^{x} + C\)
Putting x = 0, y = 1:
\(0 = e^{0} + C\)
\(C = -1\)
So the particular solution is
\(\log_e y = e^{x} - 1\)
At \(x = \log_e 2\), where \(e^{x} = 2\):
\(\log_e y = 2 - 1 = 1\)
\(y = e\)