If `inte^(-(x^(2))/(2))dx=f(x)` and the solution of the differential equation `(dy)/(dx)=1+xy` is `y=ke^((x^(2))/(2))f(x)+Ce^((x^(2))/(2)`, then the value of k is equal to (where C is the constant of integration)

Answer

To solve the problem, we need to find the value of \( k \) given the integral and the solution to the differential equation. Let's break it down step by step.

### Step 1: Understand the integral and the function \( f(x) \)
We are given that:
\[ f(x) = \int e^{-\frac{x^2}{2}} \, dx \]
This integral represents the function \( f(x) \).

### Step 2: Write the differential equation
The differential equation provided is:
\[ \frac{dy}{dx} = 1 + xy \]
This is a first-order linear differential equation.

### Step 3: Rewrite the differential equation
We can rewrite the equation in standard form:
\[ \frac{dy}{dx} - xy = 1 \]

### Step 4: Find the integrating factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int -x \, dx} = e^{-\frac{x^2}{2}} \]

### Step 5: Multiply through by the integrating factor
Multiply the entire differential equation by the integrating factor:
\[ e^{-\frac{x^2}{2}} \frac{dy}{dx} - e^{-\frac{x^2}{2}} xy = e^{-\frac{x^2}{2}} \]

### Step 6: Express the left-hand side as a derivative
The left-hand side can be expressed as:
\[ \frac{d}{dx}(y e^{-\frac{x^2}{2}}) = e^{-\frac{x^2}{2}} \]

### Step 7: Integrate both sides
Integrate both sides with respect to \( x \):
\[ y e^{-\frac{x^2}{2}} = \int e^{-\frac{x^2}{2}} \, dx + C \]
Where \( C \) is the constant of integration.

### Step 8: Solve for \( y \)
Now, solving for \( y \):
\[ y = e^{\frac{x^2}{2}} \left( \int e^{-\frac{x^2}{2}} \, dx + C \right) \]
Substituting \( f(x) \) for the integral:
\[ y = e^{\frac{x^2}{2}} (f(x) + C) \]

### Step 9: Compare with the given solution
We are given that:
\[ y = k e^{\frac{x^2}{2}} f(x) + Ce^{\frac{x^2}{2}} \]
Comparing the two expressions for \( y \):
\[ e^{\frac{x^2}{2}} (f(x) + C) = k e^{\frac{x^2}{2}} f(x) + Ce^{\frac{x^2}{2}} \]

### Step 10: Determine the value of \( k \)
From the comparison, we can see that:
- The coefficient of \( e^{\frac{x^2}{2}} f(x) \) on the left side is \( 1 \).
- The coefficient of \( e^{\frac{x^2}{2}} f(x) \) on the right side is \( k \).

Thus, we have:
\[ k = 1 \]

### Final Answer
The value of \( k \) is \( \boxed{1} \).