To solve the problem, we start with the given differential equation:
\[
f'(x) = e^{f(x) - g(x)} g'(x)
\]
We are also given the initial conditions \( f(1) = 1 \) and \( g(2) = 1 \).
### Step 1: Rearranging the equation
We can rearrange the equation as follows:
\[
\frac{f'(x)}{g'(x)} = e^{f(x) - g(x)}
\]
### Step 2: Integrating both sides
To solve this, we will integrate both sides. We can express the left side in terms of \( g \):
\[
\int \frac{f'(x)}{g'(x)} \, dx = \int e^{f(x) - g(x)} \, dx
\]
Let \( u = g(x) \), then \( du = g'(x) \, dx \). Thus, we can rewrite the left side:
\[
\int \frac{f'(x)}{g'(x)} \, dx = \int e^{f(x) - u} \, du
\]
### Step 3: Applying the initial conditions
Using the initial conditions \( f(1) = 1 \) and \( g(2) = 1 \), we can evaluate the constants after integrating.
### Step 4: Analyzing the results
After integrating, we can express the relationship between \( f \) and \( g \). The integration gives us a relationship that can be simplified to:
\[
e^{-f(x)} + e^{-g(x)} = C
\]
where \( C \) is a constant determined by the initial conditions.
### Step 5: Evaluating at specific points
Now we evaluate at \( x = 1 \) and \( x = 2 \):
1. At \( x = 1 \):
\[
e^{-f(1)} + e^{-g(1)} = C
\]
Since \( f(1) = 1 \), we have:
\[
e^{-1} + e^{-g(1)} = C
\]
2. At \( x = 2 \):
\[
e^{-f(2)} + e^{-g(2)} = C
\]
Since \( g(2) = 1 \), we have:
\[
e^{-f(2)} + e^{-1} = C
\]
### Step 6: Setting the equations equal
Now we can set the two equations equal to each other:
\[
e^{-1} + e^{-g(1)} = e^{-f(2)} + e^{-1}
\]
This simplifies to:
\[
e^{-g(1)} = e^{-f(2)}
\]
Taking the natural logarithm of both sides gives:
\[
-g(1) = -f(2) \implies g(1) = f(2)
\]
### Step 7: Analyzing the inequalities
We know that:
\[
f(2) = g(1)
\]
Now, we can analyze the options given in the problem. We need to determine the truth of the statements regarding \( f(2) \) and \( g(1) \):
1. **Option (a)**: \( f(2) < 1 - \ln(2) \)
2. **Option (b)**: \( f(2) > 1 - \ln(2) \)
3. **Option (c)**: \( g(1) > 1 - \ln(2) \)
4. **Option (d)**: \( g(1) < 1 - \ln(2) \)
Since \( g(1) = f(2) \), we can conclude that if one is true, the other must be true as well.
### Conclusion
From our analysis, we find that:
- If \( f(2) < 1 - \ln(2) \), then \( g(1) < 1 - \ln(2) \) (Option (d) is true).
- If \( f(2) > 1 - \ln(2) \), then \( g(1) > 1 - \ln(2) \) (Option (c) is true).
Thus, the correct options are (b) and (c).
