## Final Answer

## Step-by-step Solution

Problem to solve:

Group the terms of the differential equation. Move the terms of the $y$ variable to the left side, and the terms of the $x$ variable to the right side

Solve the product $2\left(y+1\right)$

Simplify the expression $2\left(y+1\right)dy$

Integrate both sides of the differential equation, the left side with respect to $y$, and the right side with respect to $x$

Expand the integral $\int\left(2y+2\right)dy$

The integral of the sum of two or more functions is equal to the sum of their integrals

Expand the integral $\int\left(4x+2\right)dx$

The integral of a constant is equal to the constant times the integral's variable

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$

Solve the integral $\int2ydy+\int2dy$ and replace the result in the differential equation

The integral of a constant is equal to the constant times the integral's variable

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $2$

Simplify the fraction $3\left(\frac{x^{3}}{3}\right)$

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$

Solve the integral $\int3x^2dx+\int4xdx+\int2dx$ and replace the result in the differential equation

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

Factor the polynomial $y^2+2y$. Add and subtract $\left(\frac{b}{2}\right)^2$, replacing $b$ by it's value $2$

Now, we can factor $y^2+2x+1$ as a squared binomial of the form $\left(x+\frac{b}{2}\right)^2$

We need to isolate the dependent variable $y$, we can do that by subtracting $-1$ from both sides of the equation

Removing the variable's exponent

We need to isolate the dependent variable $y$, we can do that by subtracting $1$ from both sides of the equation

As in the equation we have the sign $\pm$, this produces two identical equations that differ in the sign of the term $\sqrt{x^{3}+2x^2+2x+C_0+1}$. We write and solve both equations, one taking the positive sign, and the other taking the negative sign

Find the explicit solution to the differential equation