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# Solve the differential equation $\frac{dy}{dx}=\frac{2x}{3y^2}$

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##  Final Answer

$y=\sqrt[3]{x^2+C_0}$
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##  Step-by-step Solution 

Specify the solving method

1

Divide fractions $\frac{1}{\frac{1}{3y^2}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$

$3y^2=2x$
2

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

$\int3y^2dy=\int2xdx$
3

Solve the integral $\int3y^2dy$ and replace the result in the differential equation

$y^{3}=\int2xdx$
4

Solve the integral $\int2xdx$ and replace the result in the differential equation

$y^{3}=x^2+C_0$
5

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

$y=\sqrt[3]{x^2+C_0}$

##  Final Answer

$y=\sqrt[3]{x^2+C_0}$

##  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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1
2
3
4
5
6
7
8
9
0
a
b
c
d
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

### Main Topic: Differential Equations

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

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