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# Separable Differential Equation Calculator

## Get detailed solutions to your math problems with our Separable Differential Equation step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here.

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###  Difficult Problems

1

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

$y=x\frac{dy}{dx}$
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

$y\cdot dy=x\cdot dx$
3

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

$\int ydy=\int xdx$

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 $n=1$

$\frac{1}{2}y^2$
4

Solve the integral $\int ydy$ and replace the result in the differential equation

$\frac{1}{2}y^2=\int xdx$

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 $n=1$

$\frac{1}{2}x^2$

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

$\frac{1}{2}x^2+C_0$
5

Solve the integral $\int xdx$ and replace the result in the differential equation

$\frac{1}{2}y^2=\frac{1}{2}x^2+C_0$

Multiplying the fraction by $y^2$

$\frac{y^2}{2}=\frac{1}{2}x^2+C_0$

Multiplying the fraction by $x^2$

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

Combine all terms into a single fraction with $2$ as common denominator

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

We can rename $2\cdot C_0$ as other constant

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

Multiply the whole equation by $2$

$y^2=x^2+C_1$

Removing the variable's exponent

$\sqrt{y^2}=\pm \sqrt{x^2+C_1}$

Cancel exponents $2$ and $1$

$y=\pm \sqrt{x^2+C_1}$

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

$y=\sqrt{x^2+C_1},\:y=-\sqrt{x^2+C_1}$

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

$y=\sqrt{x^2+C_1},\:y=-\sqrt{x^2+C_1}$
6

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

$y=\sqrt{x^2+C_1},\:y=-\sqrt{x^2+C_1}$

##  Final answer to the problem

$y=\sqrt{x^2+C_1},\:y=-\sqrt{x^2+C_1}$

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