Differential Equations
Linear DE — Boundary Condition at Infinity
nta_pyq_2024_jan
Grade 12
Question:
Let $\alpha$ be a non-zero real number. Suppose $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function such that $f(0)=2$ and $\lim_{x\to-\infty}f(x)=1$. If $f'(x)=\alpha f(x)+3$, for all $x\in\mathbb{R}$, then $f(-\log_e 2)$ is equal to
Step-by-Step Solution
Key Concept: Rewrite as $f'(x)-\alpha f(x)=3$. I.F. $=e^{-\alpha x}$. Integrate to get $f(x)=\frac{-3}{\alpha}+ce^{\alpha x}$. Use $f(0)=2$ and $\lim_{x\to-\infty}f(x)=1$ to find $\alpha$ and $c$.
$f(x)=\frac{-3}{\alpha}+ce^{\alpha x}$. $f(0)=2$: $\frac{-3}{\alpha}+c=2$. $\lim_{x\to-\infty}f(x)=1$: requires $\alpha=-3$ and $\frac{3}{3}=1$ (satisfied), $c=1$. $f(-\ln2)=1+e^{3\ln2}=1+8=9$.
Correct Answer: 3