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Calculatrice Equations différentielles du premier ordre

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1

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

$\frac{dy}{dx}=\frac{5x^2}{4y}$
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

$4ydy=5x^2dx$
3

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

$\int4ydy=\int5x^2dx$

The integral of a function times a constant ($4$) is equal to the constant times the integral of the function

$4\int ydy$

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 $n=1$

$4\cdot \left(\frac{1}{2}\right)y^2$

Multiply the fraction and term in $4\cdot \left(\frac{1}{2}\right)y^2$

$2y^2$
4

Solve the integral $\int4ydy$ and replace the result in the differential equation

$2y^2=\int5x^2dx$

The integral of a function times a constant ($5$) is equal to the constant times the integral of the function

$5\int x^2dx$

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 $2$

$5\left(\frac{x^{3}}{3}\right)$

Simplify the fraction $5\left(\frac{x^{3}}{3}\right)$

$\frac{5}{3}x^{3}$

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

$\frac{5}{3}x^{3}+C_0$
5

Solve the integral $\int5x^2dx$ and replace the result in the differential equation

$2y^2=\frac{5}{3}x^{3}+C_0$

Multiplying the fraction by $x^{3}$

$2y^2=\frac{5x^{3}}{3}+C_0$

Combine all terms into a single fraction with $3$ as common denominator

$2y^2=\frac{5x^{3}+3\cdot C_0}{3}$

We can rename $3\cdot C_0$ as other constant

$2y^2=\frac{5x^{3}+C_1}{3}$

Divide both sides of the equation by $2$

$y^2=\frac{5x^{3}+C_1}{6}$

Removing the variable's exponent

$\sqrt{y^2}=\pm \sqrt{\frac{5x^{3}+C_1}{6}}$

Cancel exponents $2$ and $1$

$y=\pm \sqrt{\frac{5x^{3}+C_1}{6}}$

As in the equation we have the sign $\pm$, this produces two identical equations that differ in the sign of the term $\sqrt{\frac{5x^{3}+C_1}{6}}$. We write and solve both equations, one taking the positive sign, and the other taking the negative sign

$y=\sqrt{\frac{5x^{3}+C_1}{6}},\:y=-\sqrt{\frac{5x^{3}+C_1}{6}}$

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$y=\frac{\sqrt{5x^{3}+C_1}}{\sqrt{6}},\:y=-\sqrt{\frac{5x^{3}+C_1}{6}}$

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$y=\frac{\sqrt{5x^{3}+C_1}}{\sqrt{6}},\:y=-\frac{\sqrt{5x^{3}+C_1}}{\sqrt{6}}$

Multiplying the fraction by $-1$

$y=\frac{\sqrt{5x^{3}+C_1}}{\sqrt{6}},\:y=\frac{-\sqrt{5x^{3}+C_1}}{\sqrt{6}}$

Combining all solutions, the $2$ solutions of the equation are

$y=\frac{\sqrt{5x^{3}+C_1}}{\sqrt{6}},\:y=\frac{-\sqrt{5x^{3}+C_1}}{\sqrt{6}}$
6

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

$y=\frac{\sqrt{5x^{3}+C_1}}{\sqrt{6}},\:y=\frac{-\sqrt{5x^{3}+C_1}}{\sqrt{6}}$

Réponse finale au problème

$y=\frac{\sqrt{5x^{3}+C_1}}{\sqrt{6}},\:y=\frac{-\sqrt{5x^{3}+C_1}}{\sqrt{6}}$

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