Question

非齐次高阶微分方程的求解

Answer 1

特征方程 EIr^4 + k = 0, r^4 = -k/(EI) , 记为 λ = [k/(EI)]^(1/4),
得 r = λ[cos(π+2kπ)/4 + isin(π+2kπ)/4], k = 0, 1, 2, 3.
r1 = λ(1/√2 + i/√2), r2 = λ(-1/√2 + i/√2),
r3 = λ(-1/√2 - i/√2), r4 = λ(1/√2 - i/√2).
特解 y = -F/K
通解 y = C1e^(r1x) + C2e^(r2x) + C3e^(r3x) + C4e^(r4x) - F/k

Answer 2

解:微分方程为EIy^(4)+ky=-F，化为y^(4)+

ky/EI=-F/EI，设k/EI=p，-F/EI=q；再设(a+bi)^4=p，有a^4-6a²b²+b^4+(4a³b-4ab³)i=p；a^4-6a²b²+b^4=p，4a³b-4ab³=0；当p＞0时，a=±p^¼或±p^¼i，b=0；当p＜0时，a=±(-p)^¼/√2，b=±(-p)^¼/√2；方程的通解为y=Ae^[(p^¼)x]+Be^[-(p^¼)x]+Csin(xp^¼)+Dcos(xp^¼)+q (p＞0)或y=(e[(p^¼)x]+e^[-(p^¼)x])[Psin(xp^¼)+Qcos(xp^¼)]+q(A、B、C、D、P、Q为任意常数)