Differential Equations
Linear ODE — Boundary Condition at Infinity
nta_pyq_2024_apr
Grade 12
Question:
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 $9f(-\log_e3)$ is equal to ________.
Step-by-Step Solution
Key Concept: Solution: $f(x)=-\alpha/3+Ce^{3x}$. As $x\to-\infty$: $e^{3x}\to0\Rightarrow-\alpha/3=7\Rightarrow\alpha=-21$. From $f(0)=1$: $C=1-7=... $ wait: $7+C=1\Rightarrow C=-6$. So $f(x)=7-6e^{3x}$.
$f(x)=7-6e^{3x}$. $9f(-\ln3)=9(7-6/27)=9\cdot61/9=61$.
Correct Answer: 61