Solving separable differential equations by formal computations When solving a separable differential equation we move from the original equation N(y)(dy)/(dx)=M(x) to N(y)dy=M(x)dx Then we integrate both sides. My question is how precise is the expression N(y)dy=M(x)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 !

Hayley Bernard 2022-07-23 Answered
Solving separable differential equations by formal computations
When solving a separable differential equation we move from the original equation
N ( y ) d y d x = M ( x )
to
N ( y ) d y = M ( x ) d x
Then we integrate both sides. My question is how precise is the expression N ( y ) d y = M ( x ) d x ? 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 !
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Answers (1)

kartonaun
Answered 2022-07-24 Author has 14 answers
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 ( x ) so,
d y = f ( x ) d x or d y d x = f ( x )
Replacing this in the first equation we have
N ( y ) f ( x ) d x = N ( y ) d y = M ( x ) d x
as d y = f ( x ) d x
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