Step-by-step Solution

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

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 of the equality

$y^2dy=\frac{2}{3}xdx$
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 the inverse of $\frac{1}{3}$

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

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

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

Final Answer

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

Tips on how to improve your answer:

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

Related Formulas:

2. See formulas

Time to solve it:

~ 0.2 s