## Coefficients derivative

Why we always write equation in form
$$y''(x)+a(x)y'(x)+b(x)=f(x)$$

Why we never write:
$$m(x)y''(x)+a(x)y'(x)+b(x)=f(x)$$
Why we never write coefficient ##m(x)## for example?

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 Quote by matematikuvol Why we always write equation in form $$y''(x)+a(x)y'(x)+b(x)=f(x)$$ Why we never write: $$m(x)y''(x)+a(x)y'(x)+b(x)=f(x)$$ Why we never write coefficient ##m(x)## for example?
Because usually the first thing to do is divide by the coefficient of $y''$.

 But what if for some ##x##, ##m(x)=0##.

## Coefficients derivative

 Quote by matematikuvol But what if for some ##x##, ##m(x)=0##.
My guess, the behavior of the solution set changes drastically wherever m(x)=0.

 Recognitions: Gold Member Science Advisor Staff Emeritus I'm not sure how to answer your question, "Why we never write coefficient m(x) for example?" because we often do! I suspect you simply have not yet gone far enough in differential equations to see such equations. Of course, if m(x) is never 0, we can simplify by dividing by it. If m(x)= 0 for some x, that x becomes a "singular point" for the equation- either a "regular singular point" or an "irregular singular point". Regular singular points can be handled in a similar way to "Euler type" or "equi-potential equations, $ax^2y''+ bxy'+ cy= f(x)$ where each coefficient has x to the same degree as the order of the derivative. Such equations are typically approach late in a first semester differential equations class.