Step-by-step Solution

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

Go!
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
|◻|
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

Final Answer

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

Step-by-step explanation

Problem to solve:

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

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

$3y^2dy=2xdx$
2

Integrate both sides, 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

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

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

$y^{3}=x^2+C_0$
6

Removing the variable's exponent

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

Final Answer

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

Problem Analysis

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

Related formulas:

2. See formulas

Time to solve it:

~ 0.05 seconds