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Exercise

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

Step-by-step Solution

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 $x$

$\int3y^2dy=\int2xdx$
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

$y^{3}=x^2+C_0$
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 to the exercise

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

Try other ways to solve this exercise

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  • Exact Differential Equation
  • Linear Differential Equation
  • Separable Differential Equations
  • Homogeneous Differential Equation
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
  • Integrate by parts
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Similar Questions Solved

$\frac{dy}{dx}=y^2-9$

$\frac{dy}{dx}=y^2-4$

$\frac{dy}{dx}=y-y^2$

$\frac{dy}{dx}=y\left(y+2\right)$

$\frac{dy}{dx}=\left(1+e^{-x}\right)\left(y^2-1\right)$

$\frac{dy}{dx}=\frac{x}{2x-y}$

$\frac{dy}{dx}=y\left(1-y\right)$

Main Topic: Separable Differential Equations

A separable differential equation is one that can be solved by separating variables, that is, when all expressions involving a variable can be placed on one side of the equation, and expressions involving the other variable on the other side of the equation.

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