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Step-by-step Solution

Solve the differential equation $\frac{dy}{dx}=\frac{3x^2+4x+2}{2\left(y+1\right)}$

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Answer

$y^2+2y=x^{3}+2x^2+2x+C_0$

Step-by-step explanation

Problem to solve:

$\frac{dy}{dx}=\frac{3x^2+4x+2}{2\left(y+1\right)}$
1

Apply fraction cross-multiplication

$2\left(y+1\right)dy=\left(3x^2+4x+2\right)dx$
2

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

$\int2\left(y+1\right)dy=\int\left(3x^2+4x+2\right)dx$
3

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

$\int2\left(y+1\right)dy=\int3x^2dx+\int4xdx+\int2dx$
4

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

$\int2\left(y+1\right)dy=\int3x^2dx+\int4xdx+2x$
5

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

$2\int\left(y+1\right)dy=\int3x^2dx+\int4xdx+2x$
6

Expand the integral

$2\int ydy+2\int1dy=\int3x^2dx+\int4xdx+2x$
7

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

$2\int ydy+2y=\int3x^2dx+\int4xdx+2x$
8

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

$2\int ydy+2y=\int3x^2dx+4\int xdx+2x$
9

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

$1\cdot y^2+2y=\int3x^2dx+4\int xdx+2x$
10

Any expression multiplied by $1$ is equal to itself

$y^2+2y=\int3x^2dx+4\int xdx+2x$
11

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

$y^2+2y=3\int x^2dx+4\int xdx+2x$
12

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

$y^2+2y=\frac{3x^{3}}{3}+4\int xdx+2x$
13

Simplify the fraction by $3$

$y^2+2y=x^{3}+4\int xdx+2x$
14

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

$y^2+2y=x^{3}+2x^2+2x$
15

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

$y^2+2y=x^{3}+2x^2+2x+C_0$

Answer

$y^2+2y=x^{3}+2x^2+2x+C_0$