Doubt related to Linear Differential Equations

Question

Attaching Snapshot

[From Section 1.5 of the book Advanced Calculus for Applications By Francis B. Hildebrand]

I am learning differential equations, but I am having difficulty in understanding what author is trying to convey in the section enclosed inside the blue bracket?

I know $$\frac{\partial[ L{e}^{rx}]}{\partial r}=0 $$ at $r=r_1$ but I cannot follow how this leads to the conclusion for giving the solution?

Answer 1

If you differentiate the first expression in the blue box using the product rule and the chain rule the result will still have $(x-r_1)$ as a factor because $(x-r_1)$ occurs in the original with an exponent greater than $1$.