# 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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### Videos

$y=-1+\sqrt{2x^2+C_0+x^{3}+2x},\:y=-1-\sqrt{2x^2+C_0+x^{3}+2x}$

## Step-by-step Solution

Problem to solve:

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

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

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

Multiply the single term $2$ by each term of the polynomial $\left(y+1\right)$

$\left(2y+2\right)dy$
2

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

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

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

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

Expand the integral $\int\left(2y+2\right)dy$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

$\int2ydy+\int2dy=\int\left(3x^2+4x+2\right)dx$
5

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

$\int2ydy+\int2dy=\int3x^2dx+\int\left(4x+2\right)dx$
6

Expand the integral $\int\left(4x+2\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

$\int2ydy+\int2dy=\int3x^2dx+\int4xdx+\int2dx$

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

$\int2ydy+2y$

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

$2\int ydy+2y$

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 $n=1$

$y^2+2y$
7

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

$y^2+2y=\int3x^2dx+\int4xdx+\int2dx$

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

$\int3x^2dx+\int4xdx+2x$

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

$3\int x^2dx+\int4xdx+2x$

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

$3\int x^2dx+4\int xdx+2x$

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 $2$

$3\left(\frac{x^{3}}{3}\right)+4\int xdx+2x$

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

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

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 $n=1$

$x^{3}+2x^2+2x$
8

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

$y^2+2y=x^{3}+2x^2+2x$
9

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

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

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

$y^2+2y+1-1=x^{3}+2x^2+2x+C_0$

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

$\left(y+1\right)^2-1=x^{3}+2x^2+2x+C_0$

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

$\left(y+1\right)^2=x^{3}+2x^2+2x+C_0+1$

We can combine and rename $1+C_0$ as other constant of integration

$\left(y+1\right)^2=2x^2+C_0+x^{3}+2x$

Removing the variable's exponent

$y+1=\pm \sqrt{2x^2+C_0+x^{3}+2x}$

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

$y=\pm \sqrt{2x^2+C_0+x^{3}+2x}-1$

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

$y=-1+\sqrt{2x^2+C_0+x^{3}+2x},\:y=-1-\sqrt{2x^2+C_0+x^{3}+2x}$
10

Find the explicit solution to the differential equation

$y=-1+\sqrt{2x^2+C_0+x^{3}+2x},\:y=-1-\sqrt{2x^2+C_0+x^{3}+2x}$

$y=-1+\sqrt{2x^2+C_0+x^{3}+2x},\:y=-1-\sqrt{2x^2+C_0+x^{3}+2x}$
SnapXam A2

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0
a
b
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f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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

### Main topic:

Differential Equations

~ 0.36 s