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 given problem step by step, we will first analyze the differential equation and then find the value of \( k \).

### Step 1: Understand the Differential Equation
The differential equation given is:

\[ \frac{dy}{dx} = 1 + xy \]

### Step 2: Rewrite the Equation
We can rewrite this equation in standard linear form:

\[ \frac{dy}{dx} - xy = 1 \]

Here, we can identify \( p = -x \) and \( q = 1 \).

### Step 3: Find the Integrating Factor
The integrating factor \( \mu(x) \) is given by:

\[ \mu(x) = e^{\int p \, dx} = e^{\int -x \, dx} = e^{-\frac{x^2}{2}} \]

### Step 4: Multiply the Differential Equation by the Integrating Factor
Now, we multiply the entire differential equation by the integrating factor:

\[ e^{-\frac{x^2}{2}} \frac{dy}{dx} - e^{-\frac{x^2}{2}} xy = e^{-\frac{x^2}{2}} \]

This simplifies to:

\[ \frac{d}{dx} \left( y e^{-\frac{x^2}{2}} \right) = e^{-\frac{x^2}{2}} \]

### Step 5: Integrate Both Sides
Next, we integrate both sides with respect to \( x \):

\[ \int \frac{d}{dx} \left( y e^{-\frac{x^2}{2}} \right) \, dx = \int e^{-\frac{x^2}{2}} \, dx \]

This gives us:

\[ y e^{-\frac{x^2}{2}} = \int e^{-\frac{x^2}{2}} \, dx + C \]

### Step 6: Solve for \( y \)
Now, we can solve for \( y \):

\[ y = e^{\frac{x^2}{2}} \left( \int e^{-\frac{x^2}{2}} \, dx + C \right) \]

### Step 7: Relate to Given Expression
According to the problem, the solution is given as:

\[ y = k e^{\frac{x^2}{2}} f(x) + C e^{\frac{x^2}{2}} \]

Where \( f(x) = \int e^{-\frac{x^2}{2}} \, dx \).

### Step 8: Compare the Two Expressions
From our derived expression for \( y \):

\[ y = e^{\frac{x^2}{2}} \left( f(x) + C \right) \]

We can compare this with the given expression:

\[ y = k e^{\frac{x^2}{2}} f(x) + C e^{\frac{x^2}{2}} \]

### Step 9: Identify the Value of \( k \)
By comparing coefficients, we see that:

- The coefficient of \( f(x) \) in our derived expression is \( 1 \).
- The coefficient of \( f(x) \) in the given expression is \( k \).

Thus, we have:

\[ k = 1 \]

### Final Answer
The value of \( k \) is:

\[ \boxed{1} \]