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1

Solved example of First order differential equations

$\frac{dy}{dx}=x^2y$

2

Multiply both sides of the equation by $dx$

$dy=x^2ydx$

3

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

$\int1dy=\int x^2dx$

4

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

$y=\int x^2dx$

5

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

$y=\frac{x^{3}}{3}$

6

Simplifying the fraction

$y=\frac{1}{3}x^{3}$

7

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

$y=\frac{1}{3}x^{3}+C_0$

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