First of all, you can prove that exp(r 1x) and exp(r2x) are the solutions of the above-mentioned second-order homogeneous linear differential equation.

Secondly, to see whether the two roots are linearly related, we can add an x to the same root, that is, substituting xexp(rx) into the left side of the equation is also valid and a solution;

Third, the root of the yoke is that the original equation is a solution and linearly independent;

Finally, I want to say that the following other formulas are not inferences you said, but are obtained by other means like Euler formula in complex variable function. In order to be concise, the book directly gives the conclusion, and you want to prove that you can go online to find it; In fact, isn't it very simple to verify? Don't worry.