Let $y(x)$ be a function satisfying $d^2y/dx^2 - dy/dx + e^{2x} = 0.5, y(0) = 2$ and $y'(0) = 1$. If maximum value of $y(x)$ is $y(a)$. Then integral part of $(2a)$ is ____.
Step-by-Step Solution
Key Concept: Use substitution $dy/dx = t$ to reduce the second-order equation to first-order form.
Setting $dy/dx = t$, the equation $dt/dx + t e^{2x} = 0$ gives integrating factor $e^{\int e^{2x}dx} = e^{e^{2x}/2}$. Solution is $te^{-x} = -e^{2x}e^{-x} + C$, and from $y'(0) = 1$ we get $C = 2$. Thus $dy/dx = (2e^x - e^{2x})$ and integrating gives $y(x) = 2e^x - \frac{e^{2x}}{2} + \frac{1}{2}$. At $x = \log 2$, we have $y_{max} = \frac{5}{2}$, and $[2x] = [2\log 2] = [\log 4] = 1$.
Correct Answer: 1