Question

If the function \(f(x)\) satisfying the equation \([f'(x)]^{2}+4f(x)f'(x)+f^{2}(x)=0\), then \(f(x)=\)
(where \(c\) is constant of integration)

• A. \(c\cdot e^{(2\pm \sqrt{3})x}\)
• B. \(c\cdot e^{(-3\pm \sqrt{2})x}\)
• C. \(c\cdot e^{(-2\pm \sqrt{3})x}\)
• D. \(c\cdot e^{(3\pm \sqrt{2})x}\)

Hint:

Treat the equation as a quadratic in f'(x) and solve for f' in terms of f, then integrate.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The equation is a quadratic in \(f'\): \((f')^{2}+4ff'+f^{2}=0\).

Step 2: Solve for f'
\(f'=\dfrac{-4f\pm\sqrt{16f^{2}-4f^{2}}}{2}=f(-2\pm\sqrt{3})\).

Step 3: Separate Variables
\(\dfrac{f'}{f}=-2\pm\sqrt{3}\), so \(\log f=(-2\pm\sqrt{3})x+\text{constant}\).

Step 4: Solution
\(f(x)=c\cdot e^{(-2\pm\sqrt{3})x}\).

Final Answer:
The solution is \(f(x)=c\,e^{(-2\pm\sqrt{3})x}\). \[ \boxed{\text{(C) }c\cdot e^{(-2\pm \sqrt{3})x}} \]