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Calculatrice Equations différentielles séparables

Résolvez vos problèmes de mathématiques avec notre calculatrice Equations différentielles séparables étape par étape. Améliorez vos compétences en mathématiques grâce à notre longue liste de problèmes difficiles. Retrouvez tous nos calculateurs ici.

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1

Here, we show you a step-by-step solved example of separable differential equations. This solution was automatically generated by our smart calculator:

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

$3y^2dy=2xdx$
3

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 $2$

$3\left(\frac{y^{3}}{3}\right)$

Simplify the fraction $3\left(\frac{y^{3}}{3}\right)$

$y^{3}$
4

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 $n=1$

$2\cdot \left(\frac{1}{2}\right)x^2$

Multiply the fraction and term in $2\cdot \left(\frac{1}{2}\right)x^2$

$x^2$

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

$x^2+C_0$
5

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 $\frac{1}{3}$

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

Cancel exponents $3$ and $1$

$y=\sqrt[3]{x^2+C_0}$
6

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

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

Réponse finale au problème

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

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