# Step-by-step Solution

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

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### Videos

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

## Step-by-step Solution

Problem to solve:

$\frac{dy}{dx}=\frac{2x}{3y^2}$
1

Take $\frac{2}{3}$ out of the fraction

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

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

$y^2dy=\frac{2}{3}xdx$
3

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

$\int y^2dy=\int\frac{2}{3}xdx$

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 $2$

$\frac{y^{3}}{3}$
4

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

$\frac{y^{3}}{3}=\int\frac{2}{3}xdx$

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

$\frac{2}{3}\int xdx$

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 $n=1$

$\frac{1}{3}x^2$
5

Solve the integral $\int\frac{2}{3}xdx$ and replace the result in the differential equation

$\frac{y^{3}}{3}=\frac{1}{3}x^2$
6

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

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

Simplify the fraction

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

Eliminate the $\frac{1}{3}$ from the left, multiplying both sides of the equation by 

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

Solve the product $3\left(\frac{1}{3}x^2+C_0\right)$

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

We can rename $3C_0$ as other constant

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

Removing the variable's exponent

$y=\sqrt[3]{x^2+C_0}$
7

Find the explicit solution to the differential equation

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

$y=\sqrt[3]{x^2+C_0}$
SnapXam A2

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7
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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

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

### Main topic:

Differential Equations

~ 0.18 s