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Calcolatrice di Equazioni differenziali separabili

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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:

$\frac{dy}{dx}=1+0.01y^2$
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

$\frac{1}{1+0.01y^2}dy=dx$
3

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

$\int\frac{1}{1+0.01y^2}dy=\int1dx$

Solve the integral applying the substitution $u^2=\frac{y^2}{100}$. Then, take the square root of both sides, simplifying we have

$u=\frac{y}{10}$

Now, in order to rewrite $dy$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by finding the derivative of the equation above

$du=\frac{1}{10}dy$

Isolate $dy$ in the previous equation

$\frac{du}{\frac{1}{10}}=dy$

After replacing everything and simplifying, the integral results in

$10\int\frac{1}{1+u^2}du$

Solve the integral by applying the formula $\displaystyle\int\frac{x'}{x^2+a^2}dx=\frac{1}{a}\arctan\left(\frac{x}{a}\right)$

$10\arctan\left(u\right)$

Replace $u$ with the value that we assigned to it in the beginning: $\frac{y}{10}$

$10\arctan\left(\frac{y}{10}\right)$
4

Solve the integral $\int\frac{1}{1+0.01y^2}dy$ and replace the result in the differential equation

$10\arctan\left(\frac{y}{10}\right)=\int1dx$

The integral of a constant is equal to the constant times the integral's variable

$x$

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

$x+C_0$
5

Solve the integral $\int1dx$ and replace the result in the differential equation

$10\arctan\left(\frac{y}{10}\right)=x+C_0$

Divide both sides of the equation by $10$

$\arctan\left(\frac{y}{10}\right)=\frac{x+C_0}{10}$

Take the inverse of $\arctan\left(\frac{y}{10}\right)$ on both sides

$\tan\left(\arctan\left(\frac{y}{10}\right)\right)=\tan\left(\frac{x+C_0}{10}\right)$

Since arctan is the inverse function of tangent, the tangent of arctangent of $\frac{y}{10}$ is equal to $\frac{y}{10}$

$\frac{y}{10}=\tan\left(\frac{x+C_0}{10}\right)$

Multiply both sides of the equation by $10$

$y=10\tan\left(\frac{x+C_0}{10}\right)$
6

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

$y=10\tan\left(\frac{x+C_0}{10}\right)$

Risposta finale al problema

$y=10\tan\left(\frac{x+C_0}{10}\right)$

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