Differential Equations Question - None

Question

Question 21: If $\frac{f'(x)}{f(x)} = 1$ and $f(1) = 2$, then find $\left[\frac{f(3)}{2}\right]$, where $[\cdot]$ denotes the greatest integer function.

Accepted Answer

Step 1: Start with the differential equation f'(x)/f(x) = 1

Recognize that the left side is the derivative of ln(f(x)):

d/dx[ln(f(x))] = 1

Step 2: Integrate both sides with respect to x:

ln(f(x)) = x + C

Therefore: f(x) = e^(x+C) = Ae^x, where A = e^C

Step 3: Apply initial condition f(1) = 2:

2 = Ae^1

A = 2/e

Step 4: Write the explicit solution:

f(x) = (2/e)·e^x = 2e^(x-1)

Step 5: Calculate f(3):

f(3) = 2e^(3-1) = 2e^2

Step 6: Find f(3)/2:

f(3)/2 = 2e^2/2 = e^2

Step 7: Evaluate using e ≈ 2.71828:

e^2 ≈ 7.389

∴ [f(3)/2] = [e^2] = [7.389] = 7

Question:

Step-by-Step Solution

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