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 !

Question

Solving separable differential equations by formal computations
When solving a separable differential equation we move from the original equation

N (y) \frac{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 !

Answer 1

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

\frac{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

Answers (1)

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