Uploaded July 2019 | Updated September 2026, 2 weeks ago
How to solve homogeneous differential equations! This is the first video in my Dissecting Differential Equations series.
Full Dissecting Differential Equations playlist: youtube.com/playlist?list=PLug5ZIRrShJF-vOxStdHg1jyntiTKrAYO
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How to solve homogeneous differential equations! This is the first video in my Dissecting Differential Equations series.
Full Dissecting Differential Equations playlist: youtube.com/playlist?list=PLug5ZIRrShJF-vOxStdHg1jyntiTKrAYO
New math videos every Monday and Friday. Subscribe to make sure you see them!






![Integral Formula for Natural Log (without knowing the derivative)
This video proves that the natural log equals the integral from 1 to x of 1/t dt under the assumption that ln(x) is the inverse function to the exponential e^x. We can do this without already knowing the derivative of the natural log!
More details on why the integral is the inverse of e^x:
We proved in the video that any right inverse to e^x must equal that integral. However, we didnt prove that e^x has a right inverse in the first place.
We know that e^x : R → R+ is a strictly increasing function whose output can be made arbitrarily large or arbitrarily small. Therefore e^x is a bijection R → R+. Every bijective function has a two-sided inverse (see [1] below). Therefore e^x has a two-sided inverse, which in particular is a right inverse. I proved in video [2] that injective functions have at most one right inverse. Therefore the right inverse to e^x is unique if it exists. But we already know that there exists one right inverse that is also a two-sided inverse. We conclude that there exists exactly one right inverse to e^x and that this right inverse is also a two-sided inverse. Hence the integral in the video is a two-sided inverse to e^x.
[1] https://youtu.be/E-njuKKrOwg
[2]: https://youtu.be/i5ZJuJr8DwE
Calculus Problems playlist: https://youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITao
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