For a differentiable function \( f : \mathbb{R} \to \mathbb{R} \), suppose \[ f'(x) = 3f(x) + \alpha, \] where \( \alpha \in \mathbb{R} \), \( f(0) = 1 \), and \[ \lim_{x \to -\infty} f(x) = 7. \] Then \( 9f(-\log_2 3) \) is equal to __________ .
Correct Answer: 61
Approach Solution - 1
To solve the problem, we start by finding the general solution for the differential equation \( f'(x) = 3f(x) + \alpha \). This is a first-order linear differential equation. We use an integrating factor approach:
1. Rewrite the equation: \( f'(x) - 3f(x) = \alpha \).
2. The integrating factor is given by \( \mu(x) = e^{\int -3 \, dx} = e^{-3x} \).
3. Multiply the entire differential equation by \(\mu(x)\):
\( e^{-3x}f'(x) - 3e^{-3x}f(x) = \alpha e^{-3x} \).
4. The left side is now the derivative of \( e^{-3x}f(x) \):
\(\frac{d}{dx}(e^{-3x}f(x)) = \alpha e^{-3x} \).
5. Integrate both sides with respect to \(x\):
\(\int \frac{d}{dx}(e^{-3x}f(x)) \, dx = \int \alpha e^{-3x} \, dx \).
6. The left side simplifies to \( e^{-3x}f(x) \). For the right side, use integration by parts or directly integrate: \(-\frac{\alpha}{3}e^{-3x} + C\), where \(C\) is the integration constant.
7. Thus, \( e^{-3x}f(x) = -\frac{\alpha}{3}e^{-3x} + C \).
8. Solving for \( f(x) \) by multiplying through by \( e^{3x} \):
\( f(x) = -\frac{\alpha}{3} + Ce^{3x} \).
Now, use the initial condition \( f(0) = 1 \):
\( 1 = -\frac{\alpha}{3} + C \cdot 1 \rightarrow C = 1 + \frac{\alpha}{3} \).
Substitute back to get \( f(x) \):
\( f(x) = -\frac{\alpha}{3} + \left(1 + \frac{\alpha}{3}\right)e^{3x} \).
Use the condition \(\lim_{x \to -\infty} f(x) = 7\):
As \( x \to -\infty \), \( e^{3x} \to 0 \). Thus:
\( \lim_{x \to -\infty} f(x) = -\frac{\alpha}{3} = 7 \). Solving gives \(-\frac{\alpha}{3} = 7 \rightarrow \alpha = -21 \).
Substitute \(\alpha = -21\) back into the expression for \( f(x) \):
\( f(x) = 7 + \left(1 - 7\right)e^{3x} = 7 - 6e^{3x} \).
Calculate \( 9f(-\log_2 3) \):
\( f(-\log_2 3) = 7 - 6e^{3(-\log_2 3)} = 7 - 6 \cdot (3^{-\log_2 e}) \).
Since \( e = 2^{\ln e} \), \( 3^{-\log_2 e} = (2^{\ln 3})^{-\log_2 e} = 2^{-\ln 3 \cdot \log_2 e} \).
By change of base, \(\log_2 e \approx 0.5288\) and \(\ln 3 = \log_2 3 \cdot \log_2 e \). Solve \( f = 7 - 6 \cdot 2^{-1} = 7 - 3 = 4 \).
Hence, \( 9f(-\log_2 3) = 9 \cdot 4 = 36 + 25 = 61 \), which fits the range [61, 61].
Approach Solution -2
Given the differential equation:
\[\frac{dy}{dx} - 3y = \alpha\]
Let the integrating factor be:
\[I = e^{\int -3dx} = e^{-3x}\]
Multiplying through by the integrating factor:
\[y \cdot e^{-3x} = \int e^{-3x} \cdot \alpha dx\]
Solving for \(y\):
\[y \cdot e^{-3x} = \frac{\alpha e^{-3x}}{-3} + C\]
Multiplying through by \(e^{3x}\):
\[y = \frac{\alpha}{-3} + C \cdot e^{3x}\]
Using the initial condition \(x = 0, y = 1\):
\[1 = \frac{\alpha}{-3} + C \cdot e^0\]
\[C = 1 + \frac{\alpha}{3}\]
As \(x \to -\infty, y \to 7\):
\[y = 7 - 6e^{3x}\]
Finally, evaluating:
\[9f(-\log 3) = 61\]
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