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

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##  Final answer to the problem

$y=\sqrt[3]{C_0+x^2}$
Got another answer? Verify it here!

##  Step-by-step Solution 

Specify the solving method

1

Rewrite the differential equation in the standard form $M(x,y)dx+N(x,y)dy=0$

$3y^2dy-2xdx=0$
2

The differential equation $3y^2dy-2xdx=0$ is exact, since it is written in the standard form $M(x,y)dx+N(x,y)dy=0$, where $M(x,y)$ and $N(x,y)$ are the partial derivatives of a two-variable function $f(x,y)$ and they satisfy the test for exactness: $\displaystyle\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$. In other words, their second partial derivatives are equal. The general solution of the differential equation is of the form $f(x,y)=C$

$3y^2dy-2xdx=0$
3

Using the test for exactness, we check that the differential equation is exact

$0=0$
4

Integrate $M(x,y)$ with respect to $x$ to get

$-x^2+g(y)$
5

Now take the partial derivative of $-x^2$ with respect to $y$ to get

$0+g'(y)$
6

Set $3y^2$ and $0+g'(y)$ equal to each other and isolate $g'(y)$

$g'(y)=3y^2$
7

Find $g(y)$ integrating both sides

$g(y)=y^{3}$
8

We have found our $f(x,y)$ and it equals

$f(x,y)=-x^2+y^{3}$
9

Then, the solution to the differential equation is

$-x^2+y^{3}=C_0$
10

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

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

##  Final answer to the problem

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

##  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

Exact Differential EquationSeparable Differential EquationHomogeneous Differential Equation

SnapXam A2

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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.