Hayley Bernard

2022-07-23

Solving separable differential equations by formal computations
When solving a separable differential equation we move from the original equation
$N\left(y\right)\frac{dy}{dx}=M\left(x\right)$
to
$N\left(y\right)dy=M\left(x\right)dx$
Then we integrate both sides. My question is how precise is the expression $N\left(y\right)dy=M\left(x\right)dx$ ? is it a formal writing to simplify computation and get quickly into integrating both sides or is it a precise mathematical expression that has a precise mathematical meaning but goes beyond an introductory course on differential equations? Thanks for your help !

kartonaun

It is more like integrating the first equation with respect to x on both sides. And then, taking y is a function of x which it is as they both are related by this equation, let say
$y=f\left(x\right)$ so,
$dy={f}^{{}^{\prime }}\left(x\right)dx$ or $\frac{dy}{dx}={f}^{{}^{\prime }}\left(x\right)$
Replacing this in the first equation we have
$N\left(y\right){f}^{{}^{\prime }}\left(x\right)dx$ = $N\left(y\right)dy=M\left(x\right)dx$
as $dy={f}^{{}^{\prime }}\left(x\right)dx$

Do you have a similar question?