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 of the equality
Multiply the single term $2$ by each term of the polynomial $\left(y+1\right)$
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 $\int\left(2y+2\right)dy$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
The integral of the sum of two or more functions is equal to the sum of their integrals
Expand the integral $\int\left(4x+2\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
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
We can combine and rename $1+C_0$ as other constant of integration
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 $\sqrt{2x^2+C_0+x^{3}+2x}$. 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