👉 Try now NerdPal! Our new math app on iOS and Android

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

## Step-by-step Solution

Go!
Math mode
Text mode
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
|◻|
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

###  Videos

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

##  Step-by-step Solution 

Problem to solve:

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

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 Try to guess Step 3. Or become premium for the price of a latte.

The integral of a function times a constant ($-2$) is equal to the constant times the integral of the function

$-2\int xdx$

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 $n=1$

$-x^2$

Since $y$ is treated as a constant, we add a function of $y$ as constant of integration

$-x^2+g(y)$
4

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

$-x^2+g(y)$

The derivative of the constant function ($-x^2$) is equal to zero

0

The derivative of $g(y)$ is $g'(y)$

$0+g'(y)$
5

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

$0+g'(y)$
6 Try to guess Step 6. Or become premium for the price of a latte.

Integrate both sides with respect to $y$

$g=\int3y^2dy$

The integral of a function times a constant ($3$) is equal to the constant times the integral of the function

$g=3\int y^2dy$

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 $2$

$g=1y^{3}$

Any expression multiplied by $1$ is equal to itself

$g=y^{3}$
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 Try to guess Step 9. Or become premium for the price of a latte.

We need to isolate the dependent variable $y$, we can do that by simultaneously subtracting $-x^2$ from both sides of the equation

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

Removing the variable's exponent raising both sides of the equation to the power 

$y=\sqrt{x^2+C_0}$
10

Find the explicit solution to the differential equation

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

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

##  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 EquationSeparable Differential EquationHomogeneous Differential Equation

SnapXam A2

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
|◻|
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

~ 0.06 s

###  Join 500k+ students in problem solving.

##### Without automatic renewal.
Create an Account