Here, we show you a step-by-step solved example of separable differential equations. This solution was automatically generated by our smart calculator:
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
Integrate both sides of the differential equation, the left side with respect to $y$, and the right side with respect to $x$
Solve the integral applying the substitution $u^2=\frac{y^2}{100}$. Then, take the square root of both sides, simplifying we have
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
Isolate $dy$ in the previous equation
After replacing everything and simplifying, the integral results in
Solve the integral by applying the formula $\displaystyle\int\frac{x'}{x^2+a^2}dx=\frac{1}{a}\arctan\left(\frac{x}{a}\right)$
Replace $u$ with the value that we assigned to it in the beginning: $\frac{y}{10}$
Solve the integral $\int\frac{1}{1+0.01y^2}dy$ and replace the result in the differential equation
The integral of a constant is equal to the constant times the integral's variable
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Solve the integral $\int1dx$ and replace the result in the differential equation
Divide both sides of the equation by $10$
Take the inverse of $\arctan\left(\frac{y}{10}\right)$ on both sides
Since arctan is the inverse function of tangent, the tangent of arctangent of $\frac{y}{10}$ is equal to $\frac{y}{10}$
Multiply both sides of the equation by $10$
Find the explicit solution to the differential equation. We need to isolate the variable $y$
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