# 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?

pasmith
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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##.

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

My guess, the behavior of the solution set changes drastically wherever m(x)=0.

HallsofIvy
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.