微分方程式の一般解を求める

Claude3は計算が強いらしいので，微分方程式を解かせてみた。実行例に使った例題には正しい答えを返すことができました。実行例はこちらの例題を引用させていただきました。 w3e.kanazawa-it.ac.jp/math/q-and-a/bibun-eq/henkan-tex.cgi?target=/math/q-and-a/bibun-eq/bibun-eq-question1-9.html

推奨モデル - Claude

101

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微分方程式を入力してください。

y′′−10y′+29y=0

The given differential equation is:

y'' - 10y' + 29y = 0

This is a second-order linear homogeneous differential equation with constant coefficients.

To solve this equation, we can use the characteristic equation method.

The characteristic equation for this differential equation is:

r^2 - 10r + 29 = 0

Solving the characteristic equation, we get:

r1 = 5 + √(25 - 29) = 5 + √(-4) = 5 + 2i r2 = 5 - √(25 - 29) = 5 - 2i

Since the roots are complex conjugates, the general solution of the differential equation is:

y(t) = C1 * e^(5t) * cos(2t) + C2 * e^(5t) * sin(2t)

Where C1 and C2 are arbitrary constants that can be determined using the initial conditions.

This is the general solution of the given second-order linear homogeneous differential equation with constant coefficients.

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