at 3:51 you accidentally wrote constant coefficients instead of monomial coefficients. (also there is a tiny typo at 20:13 where you said -r_1 but wrote -r)
Euler equations are linear homogeneous differential equations so their solutions form a vector space. Since we are considering 2nd order equations the solution space is 2-dimensional, and so it suffices to show that the components of the solutions found in the video are linearly independent which can be done with the Wronskian (namely x^{r_1} and x^{r_2} for r_1 ≠ r_2; x^{r_1} and x^{r_1} log(x); and x^λ cos(μ log(x)) and x^λ sin(μ log(x)).)
With change of independent variable we can get equation with constant coefficients
The change of independent variable is x = e^{t}
Nice! Brings back memories of Advanced Engineering Mathematics by Kreyszig. Will you be covering the Method of Frobenius?
at 3:51 you accidentally wrote constant coefficients instead of monomial coefficients. (also there is a tiny typo at 20:13 where you said -r_1 but wrote -r)
How do we prope there are no more solutions except those in the video?
Power series. It could show that there are no other solutions.
x=e^t simplifies the general equation into a simple order 2 ordinary differential equation. We can get all the solutions from there.
Euler equations are linear homogeneous differential equations so their solutions form a vector space. Since we are considering 2nd order equations the solution space is 2-dimensional, and so it suffices to show that the components of the solutions found in the video are linearly independent which can be done with the Wronskian (namely x^{r_1} and x^{r_2} for r_1 ≠ r_2; x^{r_1} and x^{r_1} log(x); and x^λ cos(μ log(x)) and x^λ sin(μ log(x)).)