京都大学情報学研究科数理工学専攻 2023年8月実施常微分方程式

Author

Casablanca

をの多項式として次の微分方程式を考える．

をある自然数としてが解であるものとする.このとき，以下の問いに答えよ.

(i) を定めよ.

(ii) をを用いて表わせ.

(iii) 式 (1) はと線形独立な有理関数解をもたないことを示せ.

Let be polynomials of and consider the differential equation

Assume that is a solution, where is a positive integer. Answer the following questions.

(i) Determine .

(ii) Express in terms of .

(iii) Show that Eq. (1) has no rational function solution that is linearly independent of .

if , plug in,

thus

since , are both polynomials of t, has no constant term. Thus , which is in conflict with . Therefore

Let , we have

and obtain:

Let

since is a rational function, we can easily see that is a rational function and is a rational function. Let ,

if , the times of is greater than the times of . Thus , , , is the only ration function solution.