Question

Consider the concept of an integrating factor used in Problems 29-38. Are the two equations $\mathit{M}\mathit{d}\mathit{x}\mathbf{}\mathbf{+}\mathbf{}\mathit{N}\mathit{d}\mathit{y}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}$and $\mathit{\mu }\mathit{M}\mathit{d}\mathit{x}\mathbf{+}\mathit{\mu }\mathit{N}\mathit{d}\mathit{y}\mathbf{=}\mathbf{0}$ necessarily equivalent in the sense that a solution of one is also a solution of the other? Discuss.

Short answer

A solution of others is Not necessarily.

Step-by-step solution

To find the solution

They do not have to be equivalent. For example, $dx-dy=0$

It has the solution $y=x+c$.

On the other hand, $ydx-ydy=0$has $y=x+c$as a solution, but also y=0 is a solution.

However, y=0 is not a solution to the first equation.

They will be equivalent if the integrating factor is different from zero everywhere because then we could divide by it and have the following equivalences.

$\begin{array}{c}Mdx+Ndy=0\\ \iff \mu (Mdx+Ndy)=0\\ \iff \mu Mdx+\mu Ndy=0\end{array}$

Final proof

Hence, it is not necessary that $Mdx+Ndy=0$ and $\mu Mdx+\mu Ndy=0$ has an equivalent solution.