What are separable differential equations?

Question

Christopher Allers · Accepted Answer

A separable equation typically looks like:
#{dy}/{dx}={g(x)}/{f(y)}#.

By multiplying by #dx# and by #f(y)# to separate #x#'s and #y#'s,
#Rightarrow f(y)dy=g(x)dx#

By integrating both sides,
#Rightarrow int f(y)dy=int g(x)dx#,
which gives us the solution expressed implicitly:

#Rightarrow F(y)=G(x)+C#,
where #F# and #G# are antiderivatives of #f# and #g#, respectively.

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Axel Alvarez · Answer

undefined Separable differential equations are a type of ordinary differential equation that can be solved by separating the variables and integrating each side separately. These equations have the form dy/dx = f(x)g(y), where f(x) and g(y) are functions of x and y, respectively. By rearranging the equation to isolate terms involving y on one side and terms involving x on the other, you can integrate both sides with respect to their respective variables. This process typically involves multiplying both sides by dx and dividing both sides by g(y) to isolate terms involving y on one side. Then, integrating both sides with respect to x yields the solution, which can be expressed implicitly or explicitly depending on the equation and initial conditions.