# How can we divide an equation by a function, where that function isn't guaranteed to be always non-zero?

Suppose I have the following equation, where r(x) and y(x) are functions of x.

r(x)y(x) = y(x)r'(x)


My understanding is that we can divide both sides by y(x), only if we know that y(x) != 0 for all values of x.

If that's not necessarily the case, my understanding is we need to consider both cases. The case where y(x) != 0 and the case where y(x) = 0.

In the first case, we arrive at the equation r(x) = r'(x).

In the second case, where the x value makes y(x) = 0, we are unable to say anything about the relation of r(x) and r'(x).

Is my understanding correct?

If it is, than is it true that the equation r(x) = r'(x) is false? (As it's not necessarily true for all x values).

• You are correct. Jun 14, 2023 at 17:55
• Yes and the solution for $r=r'$ is $A\exp(x)$ with $A$ being a constant.
– PC1
Jun 14, 2023 at 17:56
• "We cannot conclude that $r(x) = r'(x)$" is different than "$r(x) = r'(x)$ is false." But yeah, for example imagine $y$ is the constant function $y(x) = 0$. Jun 14, 2023 at 18:25

The question's been basically answered already, but here's another way to look at it without dividing at all, if you're afraid of division (like I sometimes am). It is more formal in a way, but will not let you down if in doubt.

$$f(x)g(x)=f(x)g'(x)$$ is the same as $$f(x)\Big(g(x)-g'(x)\Big)=0$$ which is true whenever (that is, it's the same as)

$$f(x)=0\;\text{or}\;g(x)=g'(x)$$

because a product is zero when either multiple is zero.

The next step in this formal approach is to see carefully where the universal quantifier is:

$$\forall x:\big(f(x)=0\;\text{or}\;g(x)=g'(x)\big)$$

For every $$x$$, you know that either of $$f(x)=0$$ or $$g(x)=g'(x)$$ is true, but you do not know which of the two. And this is the (formal) reason you cannot say anything about $$g$$ at points where $$f(x)=0$$.

To conclude something about $$g$$ at some $$x_0$$, you need to additionally assume that the left operand of "or" is false - that is, $$f(x_0)\neq0$$. And note how we're reasoning here only pointwise - for each $$x$$ a different choice of the two conditions might be true. That's why @Jules talks about the different intervals in his answer.

You need to be careful, solve it on intervals on which $$y$$ does not vanish (you'll obviously find $$\lambda e^x$$)

• If $$y$$ vanishes only on a point $$x_0$$ but not in the neighborhood of this point, you'll have to "glue" your solutions by studying continuity on this point (you'll eventually find that the $$\lambda$$ on the left and right are the same).
• If $$y$$ vanishes $$I\neq \{x_0 \}$$, you can in this case have any thing you want for $$r$$ on this interval, as long as it is $$\mathcal{C}^1$$.
• I forgot to mention it but a complicated case is when $$y$$ vanishes on a concentration point e.g. on $$\{0,1/1,1/2,\dots,1/n,\dots\}$$ you can work with continuity to show that it is not a problem (only if this set is discrete).
• Thanks! To make the question more concrete: I'm studying a video on the Method of Integrating Factors for solving a linear 1st order ODE. In the video, in order to find the correct integration factor r(x), the teacher does the following: he divides both sides of the equation r(x)p(x)y(x) = r'(x)y(x) by y(x). I don't see him considering any case where y(x) might be 0. Any idea why that is? Jun 14, 2023 at 18:06
• Probably because $y$ does not vanish or vanishes only on a discrete set, which allows him to implicitly do the "gluing" I talked about. Be careful he might be a physicist and don't care about dividing by zero! But sometimes you can prove that $y$ does not vanish Jun 14, 2023 at 18:09
• Hmm, how can we know these assumptions about y in advance, if we're simply solving a general 1st order linear ODE? Jun 14, 2023 at 18:10
• You can't, it has to be mentioned somewhere Jun 14, 2023 at 18:11

If y(x) = 0 then you have a solution; dividing by y(x) will make it disappear.

So you can divide by y(x), find all solutions, and add the solutions of y(x) = 0.

• However, when y(x)= 0, it's correct to say that nothing can be known about the relation of r(x) and r'(x). Right? Jun 14, 2023 at 18:24