Differential Equations
Differential Equations
Allen Star Batch
Grade 12
Question:
Let $f(x)$ be a twice differentiable bounded function satisfy $2f'(x), f''(x) + 2(f'(x))^3, f''(x) = -f''(x)$. If $f(x)$ is bounded in between $y = k_1$ and $y = k_2$. Then the number of integers between $k_i$ and $k_2$ is/are (where $f(0) = f'(0) = 0$)
Step-by-Step Solution
Key Concept: Recognize the derivative structure to convert into an integrable form involving $\tan^{-1}$.
Given $2f^5(x).f'(x).[1+(f'(x))^2] = f''(x)$, we rewrite as $|f^4(x)d(f^2(x))| = d( an^{-1}(f'(x)))$. Integrating gives $\frac{f^6(x)}{3} = \tan^{-1}(f'(x)) + C$. Using $f(0) = f'(0) = 0$ yields $C = 0$. Therefore $\frac{f^6(x)}{3} = \tan^{-1}(f'(x))$, which constrains $0 \le \frac{f^6(x)}{3} < \frac{\pi}{2}$, giving $-(\frac{3\pi}{2})^{1/6} < f(x) < (\frac{3\pi}{2})^{1/6}$. The number of integers between $k_1$ and $k_2$ is 3.
Correct Answer: 3