Differential Equations
Numerical answer
Grade Class 12
Question:
<p>\\(f(x+y)=f(x)f(y)\\), \\(f(0)=1\\), \\(f'(0)=1\\). Find \\(\\int_0^{\\ln 3}f(x)\\,dx\\).</p>
<span>\(2\)</span>
<span>\(3\)</span>
<span>\(\ln 3\)</span>
<span>\(e^3-1\)</span>
Step-by-Step Solution
Key Concept: Deduce f(x) = e^x from functional equation and initial condition.
<div class='solution'><p>$f(x+y)=f(x)f(y)$ with $f(0)=1$, $f'(0)=1$ → $f(x)=e^x$. $\int_0^{\ln3}e^x\,dx=[e^x]_0^{\ln3}=3-1=2$. <strong>Answer: (1) 2</strong>.</p><p class='key-concept'>🔑 Key Concept: The only differentiable solution to $f(x+y)=f(x)f(y)$ with $f'(0)=1$ is $f(x)=e^x$.</p></div>
Correct Answer: 1