Solved example of common monomial factor
Rewrite the differential equation
Factoring by $y$
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
Integrate both sides of the differential equation, the left side with respect to
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
Expand the fraction $\frac{1-x}{x^2}$ into $2$ simpler fractions with common denominator $x^2$
Simplify the resulting fractions
Expand the integral $\int\left(\frac{1}{x^2}+\frac{-1}{x}\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $-2$
Simplify the expression inside the integral
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 $\int\frac{1-x}{x^2}dx$ and replace the result in the differential equation
Take the variable outside of the logarithm
Simplifying the logarithm
Simplify $e^{\left(\frac{1}{-x}-\ln\left(x\right)+C_0\right)}$ by applying the properties of exponents and logarithms
Simplify $e^{\left(\frac{1}{-x}+C_0\right)}$ applying properties of exponents
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Multiply the fraction and term
Find the explicit solution to the differential equation. We need to isolate the variable $y$
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