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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Final Answer

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

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

Final Answer

$y=-1+\sqrt{2x^2+C_0+x^{3}+2x},\:y=-1-\sqrt{2x^2+C_0+x^{3}+2x}$
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a
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g
m
n
u
v
w
x
y
z
.
(◻)
+
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×
◻/◻
/
÷
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

Tips on how to improve your answer:

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

Related Formulas:

4. See formulas

Time to solve it:

~ 0.36 s