Standard Exercises on Nonhomogeneous Equations and Particular Solutions

date2026-07-16document_iddoc_1be2b1476ba82eb031278a159c3a8d83description非同次線型微分方程式で同次解と特殊解を分け、未定係数法と定数変化法を使い分ける標準演習。prerequisites二階線型定数係数微分方程式の基本 / 積分法の基本type問題演習content_typeexercisestatusactive

mathdifferential-equationsexercisenonhomogeneousvariation-of-parameters

1Exercise Strategy

For a nonhomogeneous linear equation, decompose the solution as

y = y_{h} + y_{p},

where $y_{h}$ solves the homogeneous equation and $y_{p}$ is a particular solution producing the right-hand side.

2Problem 1

Solve the following differential equation by the method of undetermined coefficients:

y^{'}' - y = x .

2.1Sample Answer

○

y = C_{1} e^{x} + C_{2} e^{- x} - x .