If the solution of differential equation ` ( dy )/(dx) =1 + x +y^2 + xy ^2 ` where Y (0) =0 is ` Y= tan ( x + ( x^2)/( a))`, then a is _______

Answer

To solve the given differential equation and find the value of \( a \), we will follow these steps:

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

\[ \frac{dy}{dx} = 1 + x + y^2 + xy^2 \]

We can factor the right-hand side:

\[ \frac{dy}{dx} = (1 + x)(1 + y^2) \]

### Step 2: Separate Variables
We can separate the variables by rewriting the equation as:

\[ \frac{dy}{1 + y^2} = (1 + x)dx \]

### Step 3: Integrate Both Sides
Now we will integrate both sides. The left side integrates to:

\[ \int \frac{dy}{1 + y^2} = \tan^{-1}(y) + C_1 \]

The right side integrates to:

\[ \int (1 + x)dx = x + \frac{x^2}{2} + C_2 \]

So we have:

\[ \tan^{-1}(y) = x + \frac{x^2}{2} + C \]

where \( C = C_2 - C_1 \).

### Step 4: Solve for \( y \)
Taking the tangent of both sides, we get:

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

### Step 5: Use the Initial Condition
We are given the initial condition \( Y(0) = 0 \). Plugging in \( x = 0 \):

\[ y(0) = \tan\left(0 + 0 + C\right) = \tan(C) = 0 \]

This implies \( C = 0 \). Therefore, we have:

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

### Step 6: Compare with the Given Solution
The problem states that the solution is:

\[ Y = \tan\left(x + \frac{x^2}{a}\right) \]

From our solution, we have:

\[ \tan\left(x + \frac{x^2}{2}\right) \]

Thus, we can compare:

\[ \frac{x^2}{a} = \frac{x^2}{2} \]

### Step 7: Solve for \( a \)
From the comparison, it follows that:

\[ a = 2 \]

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

\[ \boxed{2} \]