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

Step-by-step Solution

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Final Answer

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

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

$3y^2dy=2xdx$
2

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

$\int3y^2dy=\int2xdx$

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

$3\int y^2dy$

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 $2$

$1y^{3}$

Any expression multiplied by $1$ is equal to itself

$y^{3}$
3

Solve the integral $\int3y^2dy$ and replace the result in the differential equation

$y^{3}=\int2xdx$

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 $n=1$

$1x^2$

Any expression multiplied by $1$ is equal to itself

$x^2$

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

$x^2+C_0$
4

Solve the integral $\int2xdx$ and replace the result in the differential equation

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

Removing the variable's exponent raising both sides of the equation to the power of $\frac{1}{3}$

$y=\sqrt[3]{x^2+C_0}$
5

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

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

Final Answer

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

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 EquationHomogeneous Differential Equation

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1
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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

How to improve your answer:

Related topics:

Differential Equations

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