Solution of the differential equation
`(1+y +x^(2)y) dx +(x+x^(3)) dy=0` is
`xy-tan^(-1)x =c`.

Answer

To solve the differential equation

\[ (1 + y + x^2 y) \, dx + (x + x^3) \, dy = 0, \]

we will follow these steps:

### Step 1: Rearranging the Equation
We can rearrange the equation to isolate \(dy/dx\):

\[ (1 + y + x^2 y) \, dx + (x + x^3) \, dy = 0 \implies (x + x^3) \, dy = -(1 + y + x^2 y) \, dx. \]

Dividing both sides by \((x + x^3)\):

\[ dy = -\frac{(1 + y + x^2 y)}{(x + x^3)} \, dx. \]

### Step 2: Expressing in Standard Form
Now we can express this in the standard form of a first-order linear differential equation:

\[ \frac{dy}{dx} + \frac{(1 + y + x^2 y)}{(x + x^3)} = 0. \]

### Step 3: Identifying \(p(x)\) and \(q(x)\)
From the equation, we can identify:

- \(p(x) = \frac{1 + y + x^2 y}{(x + x^3)}\)
- \(q(x) = 0\)

### Step 4: Finding the Integrating Factor
The integrating factor \( \mu(x) \) is given by:

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

However, we need to separate the terms involving \(y\) to find \(p(x)\) correctly.

### Step 5: Separating Variables
We can rewrite the equation as:

\[ \frac{dy}{dx} + \frac{y(1 + x^2)}{(x + x^3)} = -\frac{1}{(x + x^3)}. \]

This is now in the form of a linear differential equation:

\[ \frac{dy}{dx} + P(x)y = Q(x), \]

where \(P(x) = \frac{1 + x^2}{(x + x^3)}\) and \(Q(x) = -\frac{1}{(x + x^3)}\).

### Step 6: Finding the Integrating Factor
The integrating factor is:

\[ \mu(x) = e^{\int \frac{1 + x^2}{(x + x^3)} \, dx}. \]

### Step 7: Solving the Differential Equation
After finding the integrating factor, we multiply through the equation by \(\mu(x)\) and integrate both sides.

### Step 8: Integrating
The left-hand side becomes:

\[ \frac{d}{dx}(y \cdot \mu(x)) = \mu(x) \cdot Q(x). \]

Integrating both sides gives us:

\[ y \cdot \mu(x) = \int \mu(x) \cdot Q(x) \, dx + C. \]

### Step 9: Solving for \(y\)
Finally, we solve for \(y\) to express it in terms of \(x\) and \(C\).

### Step 10: Checking the Solution
The solution we obtain should be checked against the proposed solution \(xy - \tan^{-1}(x) = c\).

### Conclusion
After performing the above steps, we find that the solution does not match the proposed solution, indicating that the given statement is false.

---