d²y/dx² + dy/dx = x²Solution :d²y/dx² + dy/dx = 0The characteristic equation (特徵方程) isλ² + λ = 0,λ(λ + 1) = 0,λ₁= 0,λ₂= -1.So the characteristic solution (特徵解) isy = c₁+ c₂e^(-x)Let particular solution (令特解) bey’= ax³ + bx² + cx + dSubstituting y" into the differential equation (將特解代入原方程)：6ax + 2b + 3ax² + 2bx + c = x²a = 1/3,b = -3a = -1,c = -2b = 2So,the general solution (通解) isy = c₁+ c₂e^(-x) + (1/3)x³ - x² + 2x
d²y/dx² + dy/dx = x²Solution :d²y/dx² + dy/dx = 0The characteristic equation (特徵方程) isλ² + λ = 0,λ(λ + 1) = 0,λ₁= 0,λ₂= -1.So the characteristic solution (特徵解) isy = c₁+ c₂e^(-x)Let particular solution (令特解) bey’= ax³ + bx² + cx + dSubstituting y" into the differential equation (將特解代入原方程)：6ax + 2b + 3ax² + 2bx + c = x²a = 1/3,b = -3a = -1,c = -2b = 2So,the general solution (通解) isy = c₁+ c₂e^(-x) + (1/3)x³ - x² + 2x