## Final Answer

## Step-by-step explanation

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

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 $x$

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

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

Solve the integral $\int2\left(y+1\right)dy$ 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 $n=1$

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

Factor by the greatest common divisor $2$

Factor the polynomial $x^2+x$. Add and subtract $\left(\frac{b}{2}\right)^2$, where in this case $b$ equals $1$

Now we can factor $x^2+x+\frac{1}{4}$ as a squared binomial of the form $\left(x+\frac{b}{2}\right)^2$

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

Solve the product $2\left(\left(x+\frac{1}{2}\right)^2-\frac{1}{4}\right)$

When multiplying two powers that have the same base ($y$), you can add the exponents

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 $\sqrt{\frac{1}{2}+x^{3}+2\left(x+\frac{1}{2}\right)^2+C_0}$. We write and solve both equations, one taking the positive sign, and the other taking the negative sign