(4)/ex{-a2yJ(b)x+dx,v>-1,a>0,b>0 10.一导,球,半径为a,初温为常数,球面温度为0.求球内温度的题布和变化 11.计算积题 e-axn2 cos br lo(-)dx,其中a>0,b>0 (2) Jo(ar)Ko(B r)rdr, a>0, Re B>0. 12.高为h、半径为a的圆柱,,上下底保持温度为0,而柱面温度为ms22,求柱 内的稳定温度题布.这里取定上下底所在的平面题别为z=h和z=0 13.将下列匀数在t=0的邻域内作 Taylor端固 (1)-sin√z2+2,规定√z2+2 (2)-cos√z2-2,规定√z2-2 2 (3) sinh V2-2i,规定√2-2it=x; (4)cosh√2+2it,规定√2+2i2 14.求长圆柱形和圆形铀块的临界半径 第十八章分离变量法总结 1.将下列方程化为Stum- Liouville型方程的标准形式 (1)xd2+2a+(z+)y=0 (2)r(1 d-y 2.求解本征值问题 d +R=0 R(a)=0,R(b)=0 其中b>a>0 3.设有本征值问题 dlpez)+Ap(a)-g(z)]y=0, y(b)=aly(a)+a12y'(a), (b) (a),Wu Chong-shi 22 → ➣ ↔ ↕ (4) Z ∞ 0 exp  − a 2x 2 Jν(bx)x ν+1dx, ν > −1, a > 0, b > 0. 10. ✌❄❅❆✞❂❃☛ a ✞⑨❈☛❊❋ u0 ✞❆❇❈❉☛ 0 ✑✾ ❆❍❈❉☎❏❑✈➃➄✑ 11. ①②③❏❞ (1) Z ∞ 0 e −ax/2 sin bx I0 ax 2  dx, Z ∞ 0 e −ax/2 cos bx I0 ax 2  dx, ❣❤ a > 0, b > 0 ➙ (2) Z ∞ 0 J0(αx)K0(βx)xdx, α > 0, Re β > 0. 12. ➉☛ h ❾❂❃☛ a ☎❘❿ ❅✞✝✛● ➁➂❈❉☛ 0 ✞ ➛❿ ❇❈❉☛ u0 sin 2π h z ✞ ✾❿ ❅❍☎■✹❈❉❏❑✑➜➝➞✹✝✛●➟✒☎✫❇❏➠☛ z = h ✈ z = 0 ✑ 13. ❪✛❫❴❋✒ t = 0 ☎➡➢❍✦ Taylor ❜❝❞ (1) 1 z sin √ z 2 + 2zt ✞➤✹ √ z 2 + 2zt    t=0 = z; (2) 1 z cos √ z 2 − 2zt ✞➤✹ √ z 2 − 2zt    t=0 = z; (3) 1 z sinh √ z 2 − 2izt ✞➤✹ √ z 2 − 2izt    t=0 = z; (4) 1 z cosh √ z 2 + 2izt ✞➤✹ √ z 2 + 2izt    t=0 = z. 14. ✾ ❽❘❿④✈❘④➥➦☎➧✼❂❃✑ ♠♥➨♣ ➩➫➭➯➲➳➵ 1. ❪✛❫✶✷➄ ☛ Sturm–Liouville ➸✶✷☎➺➻④➼❞ (1) x d 2y dx 2 + 2 dy dx + (x + λ)y = 0; (2) x(1 − x) d 2y dx 2 + (a − bx) dy dx − λy = 0; (3) x d 2y dx 2 + (1 − x) dy dx + λy = 0; (4) d 2y dx 2 − 2x dy dx + 2λy = 0. 2. ✾ ✺➽➾➚✿❀❞ 1 r d dr  r dR dr  + λ r 2 R = 0, R(a) = 0, R(b) = 0, ❣❤ b > a > 0 ✑ 3. ❁✻➽➾➚✿❀ d dx h p(x) dy dx i + λρ(x) − q(x) y = 0, y(b) = α11y(a) + α12y 0 (a), y 0 (b) = α21y(a) + α22y 0 (a),