Solved example of common monomial factor
Factoring by $y$
Group the terms of the differential equation. Move the terms of the $x$ variable to the left side, and the terms of the $y$ variable to the right side
Integrate both sides, the left side with respect to $y$, and the right side with respect to $x$
We can solve the integral $\int\frac{1}{1+x}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a part of the integral with a new variable, which when substituted makes the integral easier. We see that $1+x$ 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 deriving 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, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
Substitute $u$ back with the value that we assigned to it: $1+x$
Solve the integral $\int\frac{1}{1+x}dx$ and replace the result in the differential equation
Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function
Solve the integral $\int ydy$ and replace the result in the differential equation
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Take the variable outside of the logarithm
Simplifying the logarithm
Simplify $e^{\left(\frac{1}{2}y^2+C_0\right)}$ applying properties of exponents
We need to isolate the dependent variable $x$, we can do that by subtracting $1$ from both sides of the equation
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