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自然法則の微分方程式 微分方程式の用語 一般解、初期条件、特解、指数関数、三角関数、正弦と余弦、円

微分方程式演習〈理工系の数学入門コース/演習 新装版〉 (和達 三樹(著)、矢嶋 徹(著)、岩波書店)の第1章(自然法則の微分方程式)、1-4(微分方程式の用語)、問題4の解答を求めてみる。

1

定数を求める。

y ( x 0 ) = 1 C 1 = 1

よって、特解は

y ( x ) = f ( x ) e ∫ x 0 x f ( x ' ) dx '

2

C 1 + C 2 = 0
y ' ( x ) = C 1 e x + 2 C 2 e 2 x
C 1 + 2 C 2 = - 1
C 2 = - 1 C 1 = 1
y ( x ) = e x - e 2 x

3

C 1 = y 0
y ' ( x ) = - C 1 Ω sin Ω x + C 2 Ω cos Ω x - 1 2 Ω ( cos Ω x - Ω x sin Ω x )
C 2 Ω - 1 2 Ω = 0
C 2 = 1 2 Ω 2
y ( x ) = y 0 cos Ω x + 1 2 Ω 2 sin Ω x - x cos Ω x 2 Ω

4

C 1 + C 2 log 1 = 1 C 1 = 1
y ' ( x ) = C 1 + C 2 log x + C 2
C 1 + C 2 = 2 C 2 = 1
y ( x ) = x + x log x

5

( 1 2 ) 2 + ( - 3 2 ) 2 = C 1
C 1 = 1 4 + 3 4 = 1
x 2 + y 2 = 1 y = ± 1 - x 2

初期条件より

y = - 1 - x 2

6

1 + C 1 1 - C 1 = 2 1 + C 1 = 2 - 2 C 1 C 1 = 1 3
y ( x ) = 1 + 1 3 e x 1 - 1 3 e - x = 3 + e x 3 - e - x

コード(Wolfram Language, Jupyter)

DSolveValue[{y'[x] == f[x]y[x], y[x0] == 1}, y[x], x]
Output
DSolveValue[{y''[x] -3y'[x] + 2y[x] == 0, y[0] == 0, y'[0] == -1}, y[x], x]
Output
Expand[%]
Output
Plot[%, {x, -5, 5}]
Output
DSolveValue[{y''[x] + o^2y[x] == Sin[o x], y[0] == y0, y'[0] == 0}, y[x], x]
Output
Expand[%]
Output
Simplify[%]
Output
DSolveValue[{y[x]y'[x] + x == 0, y[1/2] == -Sqrt[3] / 2}, y[x], x]

Inverse functions are being used by `1`, so some solutions may not be found; use Reduce for complete solution information.: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.



For some branches of the general solution, the given boundary conditions lead to an empty solution.: For some branches of the general solution, the given boundary conditions lead to an empty solution.
Output
Plot[%, {x, -1, 1}]
Output
DSolveValue[{y'[x] + Exp[-x] / (Exp[x] + Exp[-x]) y[x]^2 - y[x] + Exp[x] / (Exp[x] + Exp[-x]) == 0,
             y[0] == 2}, y[x], x]
Output
Simplify[%]
Output
% == (3 + Exp[x]) / (3 - Exp[-x])
Output
Simplify[%]
Output
Plot[(3 + Exp[x]) / (3 - Exp[-x]), {x, -5, 5}]
Output