微分方程式 2階微分方程式定係数2階微分方程式初期条件が付く場合の計算、特性方程式、重解、複素数解、三角関数、正弦と余弦、倍角、指数関数

学習環境

力学・電磁気学・熱力学のための基礎数学 (松下貢(著)、裳華房)の第3章(微分方程式)、3.3(2階微分方程式)、3.3.2(定係数2階微分方程式)、初期条件が付く場合の計算、問題10の解答を求めてみる。

特性方程式。

\begin{array}{l} λ^{2} - 4 λ + 3 = 0 \\ (λ - 1) (λ - 3) = 0 \\ λ = 1, 3 \end{array}

一般解。

y = c_{1} e^{x} + c_{2} e^{3 x}

\frac{dy}{dx} = c_{1} e^{x} + 3 c_{2} e^{3 x}

初期条件より、

\begin{array}{l} c_{1} + c_{2} = 1 \\ c_{1} + 3 c_{2} = - 2 \end{array}

\begin{array}{l} 2 c_{2} = - 3 \\ c_{2} = - \frac{3}{2} \\ c_{1} = \frac{5}{2} \end{array}

よって求める特解は、

y = \frac{5}{2} e^{x} - \frac{3}{2} e^{3 x}

\begin{array}{l} λ^{2} - 6 λ + 9 = 0 \\ {(λ - 3)}^{2} = 0 \\ λ = 3 \end{array}

y = (c_{1} + c_{2} x) e^{3 x}

\frac{dy}{dx} = c_{2} e^{3 x} + (c_{1} + c_{2} x) 3 e^{3 x}

\begin{array}{l} c_{1} = - 1 \\ c_{2} - 3 = 1 \\ c_{2} = 4 \end{array}

y = (- 1 + 4 x) e^{3 x}

\begin{array}{l} λ^{2} - 4 λ + 8 = 0 \\ λ = 2 \pm \sqrt{4 - 8} = 2 \pm 2 i \end{array}

y = e^{2 x} (c_{1} \cos 2 x + c_{2} \sin 2 x) = A e^{2 x} \cos (2 x + δ)

\begin{array}{l} \frac{dy}{dx} = A (2 e^{2 x} \cos (2 x + δ) - 2 e^{2 x} \sin (2 x + δ)) \\ = 2 A e^{2 x} (\cos (2 x + δ) - \sin (2 x + δ)) \end{array}

\begin{array}{l} A \cos δ = 0 \\ 2 A (\cos δ - \sin δ) = 4 \end{array}

\begin{array}{l} \cos δ = 0 \\ δ = \frac{π}{2} + n π \\ - 2 A \sin (\frac{π}{2} + n π) = 4 \\ {(- 1)}^{n} A = - 2 \\ A = - 2 {(- 1)}^{n} = 2 {(- 1)}^{n - 1} \end{array}

y = 2 {(- 1)}^{n - 1} e^{2 x} \cos (2 x + \frac{π}{2} + n π)

コード(Wolfram Language)

DSolve[{y''[x] - 4y'[x] + 3y[x] == 0, y[0] == 1, y'[0] == -2}, y[x], x]

y[x_] := 5/2Exp[x] - 3/2Exp[3x]

y[0]

y'[0]

-2

y''[x]-4y'[x]+3y[x]

Simplify[%]

y[x_] := (-1+4x)Exp[3x]

y''[x]-6y'[x]+9y[x]

Simplify[%]

y[0]

-1

y'[0]

y[x_] := 2(-1)^(n-1)Exp[2x]Cos[2x+Pi/2+n Pi]

y''[x]-4y'[x]+8y[x]

Simplify[%]

y[0]

Simplify[%, Element[n, Integers]]

Simplify[y'[0], Element[n, Integers]]