## Solving the Differential Equation: (x^(3)+x^(2)+x+1)(dy)/(dx)=2x^(2)+x, y=1 when x=0

This article will guide you through the process of solving the given differential equation, including its initial condition. We will use methods of separation of variables and integration.

### Step 1: Rearranging the equation

First, we need to rearrange the equation to separate the variables. Divide both sides by (x^(3)+x^(2)+x+1):

```
(dy)/(dx) = (2x^(2)+x)/(x^(3)+x^(2)+x+1)
```

### Step 2: Integrating both sides

Now, integrate both sides with respect to x:

```
∫dy = ∫(2x^(2)+x)/(x^(3)+x^(2)+x+1) dx
```

The left side integrates to y. For the right side, we need to use partial fraction decomposition.

### Step 3: Partial Fraction Decomposition

Factor the denominator:

```
x^(3)+x^(2)+x+1 = (x+1)(x^(2)+1)
```

Therefore, we can write the integrand as:

```
(2x^(2)+x)/(x^(3)+x^(2)+x+1) = A/(x+1) + (Bx+C)/(x^(2)+1)
```

Where A, B, and C are constants to be determined.

Multiplying both sides by (x+1)(x^(2)+1) and simplifying, we get:

```
2x^(2)+x = A(x^(2)+1) + (Bx+C)(x+1)
```

Solving for A, B, and C, we find:

- A = 1
- B = 1
- C = -1

Now, we can rewrite the integral as:

```
∫(2x^(2)+x)/(x^(3)+x^(2)+x+1) dx = ∫(1/(x+1) + (x-1)/(x^(2)+1)) dx
```

### Step 4: Solving the integral

We can now integrate the right side:

```
∫(1/(x+1) + (x-1)/(x^(2)+1)) dx = ln|x+1| + (1/2)ln(x^(2)+1) - arctan(x) + C
```

Where C is the constant of integration.

### Step 5: Applying the initial condition

We know that y = 1 when x = 0. Substituting these values into our solution, we get:

```
1 = ln|1| + (1/2)ln(1) - arctan(0) + C
```

Solving for C, we find C = 1.

### Step 6: The final solution

Therefore, the solution to the differential equation is:

```
y = ln|x+1| + (1/2)ln(x^(2)+1) - arctan(x) + 1
```

This equation describes the relationship between x and y, satisfying the given differential equation and initial condition.