[14347] Radioactive Islands (Large)

https://blog.safespot.dev/entry/14346-Radioactive-Islands-Small의 후속 문제. 

거의 모든 해설은 저기 다 있고, 일단 N 상한만 2로 늘어난 상황이기 때문에 N=2인 경우에 대해서 식을 세우면
$\displaystyle \int _{-10}^{10}\left(1+\frac{1}{x^2+\left\{y\left(x\right)-c_1\right\}^2}+\frac{1}{x^2+\left\{y\left(x\right)-c_2\right\}^2}\right)\sqrt{1+\left\{y^{\prime}\left(x\right)\right\}^2}dx$
를 최소화해야 하는 상황.

똑같이 범함수 만들면
$\displaystyle f\left(y,\ y^{\prime};\ x\right)=\left(1+\frac{1}{x^2+\left\{y-c_1\right\}^2}+\frac{1}{x^2+\left\{y-c_2\right\}^2}\right)\sqrt{1+\left\{y^{\prime}\right\}^2}$
$\displaystyle J\left[y\right]=\int _{-10}^{10}f\left(y,\ y^{\prime};\ x\right)dx$
오일러-라그랑주 방정식에 넣어 끔찍한 미분을 몇 번 하면

$\displaystyle \large{y^{\prime\prime}=\frac{\displaystyle -2\left(\left\{y^{\prime}\right\}^2+1\right)\sum _{i=1}^2\frac{y-c_i-xy^{\prime}}{\left(x^2+\left\{y-c_i\right\}^2\right)^2}}{\displaystyle 1+\sum _{i=1}^2\frac{1}{x^2+\left\{y-c_i\right\}^2}}}$

이게 뭐야...
RK4에 박으면 답 나옴... ㅎㅎ

14347번: Radioactive Islands (Large)