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

Step-by-step Solution

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Solution

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Final Answer

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

Specify the solving method

1

Divide fractions $\frac{1}{\frac{1}{2\left(y+1\right)}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$

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

Integrate both sides of the differential equation, the left side with respect to

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

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

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

Expand the integral $\int\left(3x^2+4x+2\right)dx$ into $3$ integrals using the sum rule for integrals, to then solve each integral separately

$\int2ydy+\int2dy=\int3x^2dx+\int4xdx+\int2dx$
5

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

$y^2+2y=\int3x^2dx+\int4xdx+\int2dx$
6

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

$y^2+2y=x^{3}+2x^2+2x+C_0$
7

Find the explicit solution to the differential equation. We need to isolate the variable $y$

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

Final Answer

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

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Linear Differential EquationExact Differential EquationHomogeneous Differential Equation

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Function Plot

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

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

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
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

How to improve your answer:

Similar Problems Solved

$\left(1+x^4\right)\cdot dy+x\cdot\left(1+4y^2\right)\cdot dx=0$

$\frac{dy}{dx}=\frac{2x}{3y^2}$

$\frac{dx}{dy}=y+xy$

$dx+e^{3x}dy=0$

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

$\frac{dy}{dx}=e^{3x+2y}$

$y'=x+8$

Main Topic: Differential Equations

A differential equation is a mathematical equation that relates some function with its derivatives.

Used Formulas

4. See formulas

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