I was watching a youtube tutorial on Integrating Factors and I'm lost in a part where the derivative of: xy' + 1y becomes xy.

Please I need some clarification on that part.


I was watching a another youtube tutorial on Integrating Factors and I also got lost in this part.


I don't understand how the first line converts to the other, Please I also need some clarification here.


  • 1
    $\begingroup$ Product rule.${}$ $\endgroup$ – Clayton Aug 9 '18 at 15:16
  • $\begingroup$ It's not that the "derivative of $xy'+y$ becomes $xy$", but the other way around. By the product rule, the derivative of $xy$ is $xy'+y$, which allows you to write one side of the ODE as $(xy)'$. $\endgroup$ – user170231 Aug 9 '18 at 15:19

Suppose you have


Notice that we have

this is that we have $$\frac{d}{dx}(xy)=x\frac{dy}{dx}+\frac{dx}{dx}y=xy'+y$$

by product rule.

Similarly, for $$\frac{dy}{dx}+\frac{2x}{1+x^2}y=0$$

By multiplying integrating factor of $(1+x^2)$, we have

$$(1+x^2) \frac{dy}{dx}+2xy=0$$

which can be written as

$$(1+x^2) \frac{dy}{dx}+\frac{d(1+x^2)}{dx}\cdot y$$



  • $\begingroup$ Thank you so much. This helped. $\endgroup$ – David Aug 9 '18 at 16:02

For your first question you just apply the product rule:


  • $\begingroup$ Thanks a lot. This helped. I get it now. $\endgroup$ – David Aug 9 '18 at 16:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.