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1

Solved example of Separable differential equations

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

2

Multiply both sides of the equation by $dx$

$dy=\left(y-xy\right)dx$

3

Factoring by $y$

$dy=y\left(1-x\right)dx$

4

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

$\int1dy=\int\left(1-x\right)dx$

5

The integral of a sum of two or more functions is equal to the sum of their integrals

$\int1dy=\int1dx+\int-xdx$

6

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

$y=\int1dx+\int-xdx$

7

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

$y=x+\int-xdx$

8

Take the constant out of the integral

$y=x-\int xdx$

9

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$y=x-\frac{1}{2}x^2$

10

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

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

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