Here, we show you a step-by-step solved example of separable differential equation. 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 $x$
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 $n=1$
Solve the integral $\int ydy$ and replace the result in the differential equation
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 $n=1$
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 $\int xdx$ and replace the result in the differential equation
Multiplying the fraction by $y^2$
Multiplying the fraction by $x^2$
Combine all terms into a single fraction with $2$ as common denominator
We can rename $2\cdot C_0$ as other constant
Multiply the whole equation by $2$
Removing the variable's exponent
Cancel exponents $2$ and $1$
As in the equation we have the sign $\pm$, this produces two identical equations that differ in the sign of the term $\sqrt{x^2+C_1}$. We write and solve both equations, one taking the positive sign, and the other taking the negative sign
Combining all solutions, the $2$ solutions of the equation are
Find the explicit solution to the differential equation. We need to isolate the variable $y$
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