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 Example

 Solved Problems

 Difficult Problems

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

$\frac{dy}{dx}=y\left(y+2\right)$

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

$\frac{1}{y\left(y+2\right)}dy=dx$

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

$\int\frac{1}{y\left(y+2\right)}dy=\int1dx$

Rewrite the fraction $\frac{1}{y\left(y+2\right)}$ in $2$ simpler fractions using partial fraction decomposition

$\frac{1}{2y}+\frac{-1}{2\left(y+2\right)}$

Expand the integral $\int\left(\frac{1}{2y}+\frac{-1}{2\left(y+2\right)}\right)dy$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

$\int\frac{1}{2y}dy+\int\frac{-1}{2\left(y+2\right)}dy$

Take the constant $\frac{1}{2}$ out of the integral

$\frac{1}{2}\int\frac{1}{y}dy+\int\frac{-1}{2\left(y+2\right)}dy$

Take the constant $\frac{1}{2}$ out of the integral

$\frac{1}{2}\int\frac{1}{y}dy+\frac{1}{2}\int\frac{-1}{y+2}dy$

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

$\frac{1}{2}\ln\left|y\right|+\frac{1}{2}\int\frac{-1}{y+2}dy$

Apply the formula: $\int\frac{n}{x+b}dx$$=nsign\left(x\right)\ln\left(x+b\right)+C$, where $b=2$, $x=y$ and $n=-1$

$\frac{1}{2}\ln\left|y\right|-\left(\frac{1}{2}\right)\ln\left|y+2\right|$

Multiply the fraction and term in $-\left(\frac{1}{2}\right)\ln\left|y+2\right|$

$\frac{1}{2}\ln\left|y\right|-\frac{1}{2}\ln\left|y+2\right|$

Solve the integral $\int\frac{1}{y\left(y+2\right)}dy$ and replace the result in the differential equation

$\frac{1}{2}\ln\left|y\right|-\frac{1}{2}\ln\left|y+2\right|=\int1dx$

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

$x$

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

$x+C_0$

Solve the integral $\int1dx$ and replace the result in the differential equation

$\frac{1}{2}\ln\left|y\right|-\frac{1}{2}\ln\left|y+2\right|=x+C_0$

 Final answer to the problem