如果有一长度为[tex=0.357x1.286]O1PzqaL1+AfC/NERqj1Zew==[/tex]的均匀的细棒，其周围以及两端[tex=2.357x1.286]F20DA9b5PZyvxJH27l4LOQ==[/tex]，[tex=2.143x1.286]ZWTEsSiqGBIAl5zcGo6d6A==[/tex]处均匀等到为绝热，初始温度分布为[tex=5.929x1.286]y82Ec3M62xuyzouQoYbbnJZDpDJ/UbDJHWXlR7tmgHc=[/tex]，问以后时刻的温度分布如何？且证明当[tex=1.857x1.286]G6WxJ307HB2e1l7Qz3uNbQ==[/tex]等于常数[tex=1.0x1.286]4zslcae04lcIRnIPGApodg==[/tex]时，恒有[tex=4.929x1.286]qiWnuV6XpUeYJjIHcpw8Zhb5Bz3oDL2cXR1qMH0EiFg=[/tex] .

2022-07-23

如果有一长度为[tex=0.357x1.286]O1PzqaL1+AfC/NERqj1Zew==[/tex]的均匀的细棒，其周围以及两端[tex=2.357x1.286]F20DA9b5PZyvxJH27l4LOQ==[/tex]，[tex=2.143x1.286]ZWTEsSiqGBIAl5zcGo6d6A==[/tex]处均匀等到为绝热，初始温度分布为[tex=5.929x1.286]y82Ec3M62xuyzouQoYbbnJZDpDJ/UbDJHWXlR7tmgHc=[/tex]，问以后时刻的温度分布如何？且证明当[tex=1.857x1.286]G6WxJ307HB2e1l7Qz3uNbQ==[/tex]等于常数[tex=1.0x1.286]4zslcae04lcIRnIPGApodg==[/tex]时，恒有[tex=4.929x1.286]qiWnuV6XpUeYJjIHcpw8Zhb5Bz3oDL2cXR1qMH0EiFg=[/tex] .

答案：

[b]解[/b]      即解定解问题[tex=10.286x7.214]fnpmC2J6JmQBLyo5NmGAz4xjVcZTEfpZJlZMQk2EuHVOPcv9ACLLZeAyl1cY4oFUjFAGrtUp50qbXBBPDufwtjxHm/RwK9dXTodfENJNr+6YvAIRgohAiYSu+uq4Lq9BUW09X6B9WO4puv/+GjZSLUKgt1na7u7YVvdEjxSrrncN1+H9dfsUj+J42FzRDz4eGZSjqiw5XdlvvLL9uiKrwtT3vgn3+pJ7aAbUhhrjNwMYqYM2snv66utrbhWl5WiGMUkc0BxETpVzaZywFexqv3ar3aw8ZjWq/sHzJohLFAnDK/VkcvlEYe+NnMJeIIp2nSsyBI+vd2MbTS73r//d4A==[/tex]，设[tex=5.857x1.286]RkgVFtrQZCkvBZxSsz1Xq5ZjBDS7Tjbdi6lhmrQUzuc=[/tex]代入方程及边值得[tex=13.786x3.357]fnpmC2J6JmQBLyo5NmGAz2JRpofyqyOBI2bmW1KYIseukH8lFxcQkLQ6Ksoh4PzyOsI4No+igOTHe9Umb0qO/BdJi5TKGX43g7BT/LngYttVTFoRsNAQ11KUZOswUQvCawpeBQ8M3BiOuJpQLtvIs2L2I1Lq9IXsHlgOPnM+nKB1AeH87AWHymleF9O9ef5e[/tex]，求非零解[tex=2.143x1.286]NHCnHg+iwo6BQeD+uRN+6w==[/tex]：（1）当[tex=2.429x1.071]cJXQ5yptUs5+00qZuV2Jeg==[/tex]时，通解为[tex=7.643x1.357]6Ycz2Q+XTJzkD30tqReZ60gXbWzTvMV7qBvzgk0IFJ71CxqB9KSYZ/m1zha6lGrC[/tex][tex=4.0x1.357]xKGF47wrng+Ius0JvYw+ZBKGQQLOjrQqsmHhWpCKKUs=[/tex]，[tex=10.0x1.357]5vp2Dtw6O+tGJNBcvfIXY5W7aOafUn7vjBM+DOn97IJQMhG3pChjn+vk4UQHNy8l95Sfh21JHE9aOCeq3cwtZg==[/tex][tex=6.214x1.357]O+Qen8SKyeVtGHu2+BXwLwG3mYsu0xqW6lkG0F+zaKuO1gNSWmv9vUV2EGYwn6xB[/tex]，由边值得[tex=14.786x3.357]fnpmC2J6JmQBLyo5NmGAz1i/RQIpAdXk0XQF8cDU1w2WxhjRh4Q6mOT9T1qm55mFn2MjW2xzqHG0pBcgddOHDNf3J/hgy7uce5Se22mXanlUV22Tj7R3CEXoAggINcDNAtAUsDny8N1bDNE+iltv3n0eGf7i+XJ+53RYgTY/jDswS0Qhu/Krk5ho5yg4+KiItcv3IgzofJmysmu2l5Fj+g7QV1L2hyTl6O57M3U3+QM=[/tex]，因[tex=3.929x1.286]A8gA8NdG3y6ytlF6G5zCGURkANeIecvE4zN+QcBxOyk=[/tex]故相当于[tex=10.429x3.071]fnpmC2J6JmQBLyo5NmGAz/Jw7qTzReAg3Xg3GHrBDlzTbNKC+Hp4dOxmHKWukHnpcAzxGcE5P/k+0olDELh84khgruWzoRW3osBK3x05vJfxvALzekyCtdT9SEhC2mwkx68ROc5cZUmUcEdUjLA3uA==[/tex]，视[tex=2.0x1.214]vnzjVhyzo/NIhVUgFyjLlA==[/tex]为未知数，此为一齐次线性代数方程组，要[tex=2.214x1.357]losLCuzX7/+P3RXk0RUzfQ==[/tex]非零，必需不同为零，即此齐次线性代数方程组要有非零解，由代数知必需有[tex=10.071x2.929]Uyz5s0rmQIddjb5Jc2T/YX2LvMSOLxdN98CBM7pxTig3ByIYZakZD5G58CC+ag+hzTO7kJg/TVmdahUPSYT806P4mUdr6eAUCD2xdbO5sKxIeKVKJEss9WvGyVQaygzVnQZ0gIA2+P/Neoc2kC+RgQ==[/tex]，但[tex=9.571x2.929]Uyz5s0rmQIddjb5Jc2T/YX2LvMSOLxdN98CBM7pxTig3ByIYZakZD5G58CC+ag+hz3I6LPI8cIS1s9anxDorLdhkjTlM0FXj/SRCq2I/Znhc2j4KzXMxuKssXDSHcPHOdOdGAzFy/b/4IXfraW7xpQ==[/tex][tex=8.5x1.357]3n9VhpZmJWuN5m4ntqgJ+l4g9BikWSzygCsAPdAzaGA+/KjY1DCWA/X+BXe7ycLnto7Mn7C6fWwn1eT0kc+ZIA==[/tex]，因[tex=2.143x1.071]qa916B8X2XrfiRwAsb4JXA==[/tex]，[tex=3.929x1.286]A8gA8NdG3y6ytlF6G5zCGUrRUM/h4imTi9uQuIOGljM=[/tex]，[tex=0.929x1.0]QJFhrTgZT3zbRAftoEeV7A==[/tex]为单调增函数之故 . 因此没有非零解[tex=2.214x1.357]losLCuzX7/+P3RXk0RUzfQ==[/tex] . （2）当[tex=2.357x1.286]vQoqjVxMQSKqWEw5yAg5YA==[/tex]时，通解为[tex=6.214x1.286]OOai7PD3soszUFB8hSLTxQ==[/tex]，[tex=4.286x1.286]5vp2Dtw6O+tGJNBcvfIXY+lcbntJORj6xtCGsjt9WL4=[/tex]，[tex=9.5x1.286]azMjdRXha0iuMW5gG3W1qLk1xRXsQvjn2sA5oblncBldXWsbhUGhVWdkTFmbmqYn[/tex]，即[tex=0.5x1.286]PGyKeLDo0qv9T0n29ldi6w==[/tex]可任意，故[tex=3.929x1.286]jFEKRo4wqxcax6gdZYdCRfOMEprZLCh3+d+hESzVBt4=[/tex]为一非零解 . （3）[tex=2.357x1.286]aKg49BUpv3BWm3erigiDBw==[/tex]时，通解为[tex=8.5x1.286]cw2oICe+wL+QJD+vyZDI2TUHd2Jx/Wrwv01davxZR+UV8iQvZGVw+x4XUan8A3oo[/tex][tex=4.286x1.286]7VkzAsuNo4izjJ1U9M+flXJycCWscx3DFlrA6F/ccTE=[/tex]，[tex=10.786x1.286]5vp2Dtw6O+tGJNBcvfIXY0S5KIKxkxqT/ZEMWNpZrrhBOtASx+EsTZBfppfHo7IN1gr1ePl77z+wxfpA0EYiSw==[/tex][tex=5.714x1.286]M7t+RvD+Db6czYgkEWzS95TL8IsOJDXRuaJ/nrXumnB34b4W+RyMmkWR5rhI9BdC[/tex]，由边值得[tex=18.857x3.643]fnpmC2J6JmQBLyo5NmGAz/S209FOlWx2xXRnhmbGjxla03hEViJ7LAVvXnH6t9a/sEgj39JIOxPaHqB8AsjTQAOPh9QdBmjiWNYyJqFUBm1+xcBGGbyenxv+AFcwhiDyyA4og06jHVFa8T7ynMo7bmknSXZi3MKqEytA6r3gduYtRLTjafDWht1dZbwgCstT3e6Ys5D0NqcsQnsaVaAgC96/RKtz2z7uTy4JEeelvZE=[/tex]，因[tex=3.214x1.286]Wi1cVPAClemTsOHMq1+b14hNQTS2b85lhhH01rrWEcM=[/tex]，故相当于[tex=7.143x2.786]fnpmC2J6JmQBLyo5NmGAz9KQDAMBP9SoGUxnIbV0I0atBQ2p37FcyNLedakN9sayBUV1fg4FctXCY2J2DHYKtXGoR4eCyNcj1LOiIAsBVgY=[/tex]，要[tex=2.214x1.357]yPAaCZ94sFkc35i4vfRXBQ==[/tex]非零，必需[tex=2.571x1.286]5sMsKFV570Hy4WENcd4qjg==[/tex]，因此必需[tex=4.5x1.357]cRs3sbNfbd9gmo54aib7NwagmjtHapBIYtpc9pSErjc=[/tex]，即[tex=4.214x1.286]Wi1cVPAClemTsOHMq1+b15agUA0Gxh6fpg7zT/sOSUs=[/tex]（[tex=0.643x1.286]ZsZs11iKEvfmzDIurZth8g==[/tex]正数），[tex=4.0x1.786]Wi1cVPAClemTsOHMq1+b1y3J2NQT7Axlx3dIlB4+WYiIcMYg62XLVh0mYy7sunyc[/tex] （[tex=0.643x1.286]ZsZs11iKEvfmzDIurZth8g==[/tex]正数），这时对应[tex=6.929x1.786]CTPQCV8aXR6H3wIoHR1yQ0mqqLS8qnugxItRb5x5O5r3somU45PA2+QwrMugVw4P[/tex]（取[tex=2.5x1.286]jjBTsydws72GowVhIvqtLw==[/tex]），因[tex=0.643x1.286]ZsZs11iKEvfmzDIurZth8g==[/tex]取正整数与负整数对应[tex=2.214x1.357]losLCuzX7/+P3RXk0RUzfQ==[/tex]一样，故可取[tex=4.0x1.786]Wi1cVPAClemTsOHMq1+b1y3J2NQT7Axlx3dIlB4+WYiIcMYg62XLVh0mYy7sunyc[/tex]，[tex=4.857x2.0]Xr89GRH5BmFr8n2WaNjOzNi8A5MijuyR5kThZ3j8Mrpa32vRDeSB6+Wfn+j6pOap[/tex]，[tex=4.857x1.286]YPiXln35VbFZwRw1qebPNC/mxvz5Wi8NCIeJXzYWeYc=[/tex]，[tex=7.357x1.786]XBBXzlKIYWqVmsG5iaJaluv0dTQWrul0JCnQ6pbCnkoi6BRgrkpBq7wQWF5ZujJN[/tex]，[tex=4.857x1.286]YPiXln35VbFZwRw1qebPNC/mxvz5Wi8NCIeJXzYWeYc=[/tex]，对应于[tex=2.357x1.286]vQoqjVxMQSKqWEw5yAg5YA==[/tex]，[tex=4.286x1.286]qvU7ge2VQ2cmvDf9c2vHGw==[/tex]，解[tex=0.643x1.0]iollMFTzm3iqFEHRyKQe1A==[/tex]得[tex=3.929x1.357]ah02vPwR0gsvEh8w5uDTP8dGsUCyjspQNibfikpw/4Q=[/tex]，对应于 [tex=4.857x2.0]Xr89GRH5BmFr8n2WaNjOzNi8A5MijuyR5kThZ3j8Mrpa32vRDeSB6+Wfn+j6pOap[/tex]，[tex=7.357x1.786]XBBXzlKIYWqVmsG5iaJaluv0dTQWrul0JCnQ6pbCnkoi6BRgrkpBq7wQWF5ZujJN[/tex]，解[tex=0.643x1.0]iollMFTzm3iqFEHRyKQe1A==[/tex]得[tex=8.714x1.5]nKEkGyp7T2h3Y+TxIYrSbHxscALf3eoBlp6ARwQpkKjpmjmDQGuzhT1WXSGRi0UUDNqy64AoPWoRtgN1Qpzf6A==[/tex]，由迭加性质，解为[tex=5.857x1.286]9zKlgnB3bKKFu5IlvRCKUvdCZbZBe1vB424mxlB5uYs=[/tex][tex=11.429x2.714]ySadpvq7BrEZCGdcnD6+aRCHEZyyKHfLogAA+NcqEgwOoUORhpx6SXuvQAVA7r4W02m/OX/NGgD9/q41elFkc+umUg8BBt6U0bPkXuze5ukFzsJC8CXGJotBVwlITzc2Mfv/VZNEW2dQZhhPvDksFw==[/tex][tex=11.0x2.714]LCs/jzl+nr3KBTJXBn4IiT8wElDucy56fy/Xemvbl7TkKL/H7eSr8sqJY24zMgAtxVkpVP1bSfB1ocUkVZVEJMwQwHQQhTbNQ0vVCsmF3+Z57AmmD02/gdpy6nzzu4LEiNff1pKAtUYSCLUXW5y1Pg==[/tex] . 由始值得[tex=9.643x2.714]/lM7lvY4K2XfzrATaOOR5/HCTgd/OLD+F0ArbTEgAknvvJJ5AMgwfgtBsYjErQ4W2ZHouAYBjBk7Xwg+1l1RpA==[/tex]，因此[tex=7.714x2.5]XtIY4L119eAQJnMr6/+TIlA8ksX663AVmb9jXm6SNo5j+Zv6JgSrCO65AZhvOsnW[/tex]，[tex=11.357x2.5]sLeH6gcuzCTm8ne7hPB09TQisKlsopNg2N1AkSS9A/b95BUPkUQlZO95syDRHE0WLORbdAxyWGknslAbxl1Qpr/VQ994DvrS8qDEay+yZ9o=[/tex]，[tex=4.857x1.286]YPiXln35VbFZwRw1qebPNC/mxvz5Wi8NCIeJXzYWeYc=[/tex]，所以[tex=10.0x2.5]+LMzEFQF6Nu3PFg+nZ44vNId+sje9KW+NEjCITbm/TUE8V2vlJV6GZv6JlTVjmc2[/tex][tex=5.714x2.714]ySadpvq7BrEZCGdcnD6+aW9DXU5f3FG7K/yhFwrwR0Vfm4/2Vg6oFZjAN0qQDgQOSi8civZT+A8/Wk+eb71QlQ==[/tex][tex=8.571x1.857]rHg/iD8KFjq/bCE0e4I0eI0kjzZrM6ikIRvNSkTtLUrHg98tGJDaQ88Frh7O9TnAOUPB7sFdctMVTjVcmH8Ka7n08rt/Iqte/IY5Ddjdei5/jQTyDSQQu1m/V76njfVc[/tex][tex=3.429x1.786]rHg/iD8KFjq/bCE0e4I0eAPWm0AA2VuIWdxvVzskJ5s=[/tex]，当[tex=7.786x1.286]fKKVhsexDWuBiI4czbKJG3EobIH6iLDatfMD+poIFgU=[/tex]时，[tex=2.143x1.286]0yjOrQElUlKLfn2ZUYr3dg==[/tex][tex=6.643x2.5]96S6bMipU1ogl0Af7bm6d49dJHVBtNReEnSQ/+LGZgH3sRez09zkTdEi0ZZH8XiM[/tex]，[tex=2.286x1.286]/6rjMMDxr6eprPEhhVzQgg==[/tex][tex=9.714x2.5]QJ0zN8lu0/QncNMyn8YHH+8T1D1Fkll2zhKGF0ZYP8ex4YBSUtjTifmWqsFIXV+RoEaSXc8XMLBEPMoickPjkq7YQ3BLbe+k6J/6LNTog14=[/tex]，[tex=4.857x1.286]YPiXln35VbFZwRw1qebPNC/mxvz5Wi8NCIeJXzYWeYc=[/tex]，所以[tex=4.429x1.357]5kUeQmSDEHqPrJg3FMjMfg==[/tex] . 

举一反三

内容

0
某人对商品x的需求函数是[tex=5.214x1.214]0m6eBd5eyK0NjuxeKfwtIw==[/tex]，[tex=4.214x1.214]I717YsPbj8Rnym1v2XQ+sFNkUl7mqUsGwbjwjXmy2xc=[/tex]，这里[tex=0.571x1.0]Za328cIB4SeR7rrzY+MM5Q==[/tex]是[tex=0.571x0.786]ZSLOI4fiO1oAbVC5M8IVkA==[/tex]的价格。如果商品x 的价格是0.5元，那么他对商品x的需求价格弹性是未知类型：{'options': ['-10', '- 1/5', '-1/10', '\xa0- 1/3'], 'type': 102}
1
判断下列命题是否为真：（1）[tex=3.643x1.357]/5abqJjwKZ1qr+6hsVFF5EBvfq3ggOFNlHMClz0h9nk=[/tex]（2）[tex=2.929x1.357]rGJpyjIjJpbcoBTWxP0Jiw==[/tex]（3）[tex=4.5x1.357]2wycHMoqU83MyEp17iBils58bR7YLuCTI2G9NVAdlfY=[/tex]（4）[tex=5.214x1.357]CTz2gu+IIm1GgNmYMGaduCRtA41wnW4WqwRWwEhq6aA=[/tex]（5）[tex=4.857x1.357]1DcE2BMMOaZhTuxR/mjgsboXxfg5ET59Dp4I/jjEDuw=[/tex]（6）[tex=4.643x1.357]BSryrsQYOvTP2hTWRu6t4nAuJwlSs4L9jaq70EpB+Us=[/tex]（7）若[tex=6.0x1.357]y0IZLUnBO88nR8WBZYvd7QXv5S1OMINV5cQNzPyiyAc=[/tex]，则[tex=3.429x1.357]1brfPwTkVVIX4GfoMIUskA==[/tex]（8）若[tex=7.643x1.357]MhLfJXZnhbXiB0x3oNtFzThV4Y1mJxe1VYr7PkJE/T6hmTD3WWp+UxbNwvUQ6DHk[/tex]，则[tex=4.143x1.357]LZUA94ISo1po5HWsOVeBCjo0rMvj7uw3bGw5HiZenrI=[/tex]
2
对于以下两种情形:(1)x为自变量,(2)x为中间变量,求函数[tex=2.214x1.214]sy9gaFRMGlrH59gm9bWSDg==[/tex]的[tex=1.5x1.429]5W5tOYbJ+LlsRP2dMsi4byxwtjvvL/3u7NEzPV5PWp0=[/tex]
3
一根长为[tex=0.357x1.286]O1PzqaL1+AfC/NERqj1Zew==[/tex]的细杆表面绝缘，其初始温度分布如图所示，由[tex=2.143x1.286]5u+Kfi2y78D0EeH+RhjPUA==[/tex]开始两端温度保持于[tex=1.643x1.286]PiWgBncJOtvCPa+v7LcA9UtwFGSiOCdfwZ5/0Nj81ko=[/tex]，求杆上温度分布.[img=481x298]178afbeeb329949.png[/img]
4
下列函数是哪些函数复合而成的？（1）[tex=4.214x1.286]6PuLCl/TwscTl61WSePGog==[/tex]；（2）[tex=5.214x1.286]+mZ2Cm2OprRKGTGg0iqmyZx+4lZ796PxrSQNx30R9UU=[/tex]；（3）[tex=4.214x1.357]jTbrMH55vzOFOJlLSnfh103OHFmRhIjXZGzPnfweOX0=[/tex]；（4）[tex=6.071x1.286]W2A0mViHY0pK74wEByr6ED5K+AKV/pxHaeQdYGQBxwc=[/tex]；（5）[tex=6.714x1.429]8up/G1s+GteD9ejcGkFVmYl3TTtTik5kuwrPDCv0JkbGIWyY33cnaw7XtBiPcSnh[/tex]；（6）[tex=5.714x1.286]APaFs2rWyubdkzLcUVVxVJSSAsLEOtXn4KjnToE2BQA=[/tex]；