Here, we show you a step-by-step solved example of separable differential equations. This solution was automatically generated by our smart calculator:
Factoring by $y$
Grouping the terms of the differential equation
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
Factor the polynomial $x^2+x$ by it's greatest common factor (GCF): $x$
Simplify the expression $\frac{-\left(2x-1\right)}{x^2+x}dx$
Integrate both sides of the differential equation, the left side with respect to $y$, and the right side with respect to $x$
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
Solve the integral $\int\frac{1}{y}dy$ and replace the result in the differential equation
Take out the constant $-1$ from the integral
Rewrite the fraction $\frac{2x-1}{x\left(x+1\right)}$ in $2$ simpler fractions using partial fraction decomposition
Expand the integral $\int\left(\frac{-1}{x}+\frac{3}{x+1}\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
We can solve the integral $\int\frac{3}{x+1}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $x+1$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by finding the derivative of the equation above
Substituting $u$ and $dx$ in the integral and simplify
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
Multiply $-1$ times $-1$
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
Multiply $-1$ times $3$
Replace $u$ with the value that we assigned to it in the beginning: $x+1$
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 $\int\frac{-\left(2x-1\right)}{x\left(x+1\right)}dx$ and replace the result in the differential equation
Take the variable outside of the logarithm
Simplifying the logarithm
Simplify $e^{\left(\ln\left(x\right)-3\ln\left(x+1\right)+C_0\right)}$ by applying the properties of exponents and logarithms
Simplify $e^{\left(-3\ln\left(x+1\right)+C_0\right)}$ by applying the properties of exponents and logarithms
We can rename $e^{C_0}$ as other constant
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Multiply the fraction by the term
Find the explicit solution to the differential equation. We need to isolate the variable $y$
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