## 2019年9月27日金曜日

### 数学 - Python - 微分積分学 - 微分法の公式 - 無理関数 - 多項式、代数方程式、累乗根(平方根、立方根)

1. $\begin{array}{l}y=\sqrt{1+{x}^{3}}+\sqrt[3]{1+{x}^{3}}\\ y-\sqrt{1+{x}^{3}}=\sqrt[3]{1+{x}^{3}}\\ {\left(y-\sqrt{1+{x}^{3}}\right)}^{3}=1+{x}^{3}\\ {y}^{3}-3{y}^{2}\sqrt{1+{x}^{3}}+3y\left(1+{x}^{3}\right)-\left(1+{x}^{3}\right)\sqrt{1+{x}^{3}}=1+{x}^{3}\\ {y}^{3}+3y\left(1+{x}^{3}\right)-\left(1+{x}^{3}\right)=\left(3{y}^{2}+\left(1+{x}^{3}\right)\right)\sqrt{1+{x}^{3}}\\ {\left({y}^{3}+3y\left(1+{x}^{3}\right)-\left(1+{x}^{3}\right)\right)}^{2}={\left(3{y}^{2}+\left(1+{x}^{3}\right)\right)}^{2}\left(1+{x}^{3}\right)\\ {y}^{6}+2{y}^{3}\left(3y\left(1+{x}^{3}\right)-\left(1+{x}^{3}\right)\right)+9{y}^{2}{\left(1+{x}^{3}\right)}^{2}-6y{\left(1+x\right)}^{2}+{\left(1+{x}^{3}\right)}^{2}\\ =\left(9{y}^{4}+6{y}^{2}\left(1+{x}^{3}\right)+{\left(1+{x}^{3}\right)}^{2}\right)\left(1+{x}^{3}\right)\\ {y}^{6}-3\left(1+{x}^{3}\right){y}^{4}-2\left(1+{x}^{3}\right){y}^{3}+3{\left(1+{x}^{3}\right)}^{2}{y}^{2}-6{\left(1+{x}^{2}\right)}^{2}y-{\left(1+{x}^{3}\right)}^{2}{x}^{3}=0\end{array}$

よって、 （4）の形の代数方程式を満足する。

コード

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, sqrt, root, solve

print('1.')

x, y = symbols('x, y')
cs = [1,
0,
-3 * (1 + x ** 3),
-2 * (1 + x ** 3),
3 * (1 + x ** 3) ** 2,
-6 * (1 + x ** 3) ** 2,
-(1 + x ** 3) ** 2 * x ** 3]
eq = sum([c * y ** (6 - i) for i, c in enumerate(cs)])
y0 = sqrt(1 + x ** 3) + root(1 + x ** 3, 3)

for o in [eq, solve(eq, y), eq.subs({y: y0}).simplify()]:
pprint(o)
print()

$./sample1.py 1. 2 2 3 ⎛ 3 ⎞ 6 4 ⎛ 3 ⎞ 3 ⎛ 3 ⎞ 2 ⎛ 3 ⎞ - x ⋅⎝x + 1⎠ + y + y ⋅⎝- 3⋅x - 3⎠ + y ⋅⎝- 2⋅x - 2⎠ + 3⋅y ⋅⎝x + 1⎠ - 6⋅y 2 ⎛ 3 ⎞ ⋅⎝x + 1⎠ [] 0$