试讨论微分方程 [tex=2.643x1.357]rjzw0bBUODiY66l+Mq83xFnaQ6h+DDYcnoJTwK2KtQE=[/tex] [tex=5.929x1.357]c3/wK/fZIQMVyYGgjFtrJQ==[/tex] [tex=6.143x1.429]12OLMovPZNDq9F/bksroMHEKYtpM5TrKo+Jfn6/PPOY=[/tex] 边值问题的可解性并求解

2022-06-19

试讨论微分方程 [tex=2.643x1.357]rjzw0bBUODiY66l+Mq83xFnaQ6h+DDYcnoJTwK2KtQE=[/tex] [tex=5.929x1.357]c3/wK/fZIQMVyYGgjFtrJQ==[/tex] [tex=6.143x1.429]12OLMovPZNDq9F/bksroMHEKYtpM5TrKo+Jfn6/PPOY=[/tex] 边值问题的可解性并求解

答案：

解：非齐次方程有特解 [tex=4.929x2.357]eu86rm5EWQQQePTVKwD67W3CuMQeHy9DKk+Br+VQGWpVAUNzi0vkJGIaDNHmcpNA[/tex] 对应的齐次方程有基本解组 [tex=3.786x1.357]P3FjwFRtvDfrArKB2u4sgA==[/tex] [tex=3.857x1.357]XT8jHRyxJKZW8kjrT1/qMQ==[/tex] 边值条件对应 [tex=16.429x1.429]/pn8RHrwrY08JdGA35CbkEzyoZX7za3H7LmWUF0vuLqOIb/wayjBErdphpkEIRTEKKBzDsTK7Kpu4eV7dWWsaFvFqb3kUtdt/a5vMcUn3+c=[/tex] 于是 [tex=22.0x5.214]hFGPqaTLJbz58rkObOard3xCdy0OyslvfX30XfGK5YmqyRHQNBH3UBG+o4vju9UBO4n/9bY8cZJt6cgRSRsHL6bLUIGJDeB5Cl+Xzd4GbwKLYpQLLlAgcCTjBJY4y+U+n+pUv9DEYuMOYIrhSTb/MMYrIVKqMfzVP/IZRvo2T1yQhAiO8U2FZre+rb++zoB/Vroe9KubGrI2SmX5YG+IEBzH90uJpW5yRru4t37MbiC2lWabo1LbI7qKjwcv5CiwgyjaBeV/hPto+kkVUaOK80ai2xTdB45B6TX+/61Z5Oj5RwCROF4r3QP028H2h26pYJlexwksSEGebP49jeH+ilroaUQ620GCJOAJXgxkdLi4BhlJks/3zK4LGthvqt1AG3AZgkArOR/iT2HCKo4nrQ==[/tex]其秩为 2 与矩阵 [tex=14.357x3.357]hFGPqaTLJbz58rkObOard1ibK+NmenREtPXA/+WQHSKcbNzNWcbdyAmJ/R/EKTseyPGa2yCtIbiVz9nVpeMQZGM9NOo6sQxStAZ5GM+YQ8aQDR3BMLihGt2ieyxd3wLulYxVbOZ6+OjiWB2cmfc+Z1JRR9iPGT5/FblC5b8TiSMkP373/qklCcysNKhZJwt4nrLpCbscqrWGB88xWnJTur+lumYmNs+jWu5920UrELhhagdz94bsZDeUdiJOmFhKwp17nscSKzwdpvleBagOx+bscy9WQzO/IEpgV7Kpbow=[/tex]的秩 1 不同. 边值问题不可解. 事实上, 非齐次方程有通解 [tex=8.357x2.357]Bj+yZNVE8lWMqIzcemX+RlwJ5fgBnc8PFEmx8P3lKN3UqNzRdTz+uKIkdiziG7E4[/tex] 由边值条件应有 [tex=5.0x2.357]85rydfRyr1YmOGFeJNXIG/T4hpQEP0ABpS8pZgxfP4o=[/tex] [tex=4.429x2.357]FLHhHpcvwKm4xcTPntqgAeYUaMXQX5dhBYNT0D/GioQ=[/tex] 得 [tex=3.714x2.357]EtFBJp7wk2arhII2mnrGUNsEdwPTZTqr+3YzwinnTTw=[/tex] [tex=3.714x2.357]LvUloUuzAwLH5h6XwNhGk89S8Yy8LZDWE1Ed8ufkrZQ=[/tex] 相互矛盾. 故无法求出满足边值条件的常'数 [tex=2.357x1.0]qD/TCRGKObKddYrUdUHFoJAUYVsStbcpGAohKjdovhI=[/tex]

举一反三

内容

0
给出微分方程 [tex=4.357x1.357]rjzw0bBUODiY66l+Mq83xNVKTdA+2lP5AAn2W7hb/fU=[/tex] [tex=3.357x1.357]8DYd2RqYxE+ARDOwacsTEg==[/tex] [tex=4.429x1.357]UWJRw4aJIR+mLOLcIubEpg==[/tex] 边值问题的解
1
已知线性微分方程的一个解，试求方程的另一个无关解，并求出方程的通解：[tex=5.214x1.286]7cBtLMZiCvN+oN5Ldx4kEHgzPTnrMWOVYUl5BL4DP8o=[/tex][tex=8.214x1.286]lA9GrrrRvn8fK0q7V8Ret2xRLYRUDVREPgZ53uEv8AY=[/tex]，[tex=3.143x1.286]DiBqXIr0m0AmdLcNnhoTkaVgrpA6cECbwsZKgJwKfC4=[/tex] .
2
设[tex=5.929x1.357]y5rqX7t57E9WfcJVLDLZLDwp5zGkfbdsAmzRs7wX8aA=[/tex]与[tex=0.929x0.786]D9maNLyVVGrC3QbL9jjRWg==[/tex]是正整数，又设[tex=7.429x1.5]olhLc0i0mC1kBiE23pYeDtefGdBixaHlDI+/EwbYT+g=[/tex]，证明同余方程[tex=7.143x1.5]0nSi38QdKl3+Z+4EotXG+TSacA6eKuqs4ipGkq+KA4U=[/tex]的一切解x都可以表示成[tex=7.571x1.357]DmiJBMaau3TV3IEe47EFZ7m+atfQ0yOLHaTjl5Zs0Uc=[/tex]，其中y满足同余方程[tex=7.0x1.5]E0NW/9Kd44LtfVknhF40AC8iVoxsILps/RBhbBSnAmk=[/tex]
3
试将微分方程组 [tex=16.357x1.429]TUbuj6qHKsvryc3kNRMLKPYVMiMeUm8vr6XKvpuPvpu+wdbB/VU/m2viIPXbOJr+WYFw1Z7tGoY56OaKzYtXZg==[/tex]化为等价的微分方程, 并求出方程的解.
4
求解线性规划问题:[tex=11.643x6.714]IY0RY5AQWEEETIe2QnDJjTHY+qT4CpWLe3kVva/jfn/TuazdhSu3XTizX9vOHQ9dzt50jOvEzCsmE50UzwuSTQrNX041/CBGE2TQN6XyYWkVjH7DuKmRF8OXb34+l6lQi4bUOXEgkcrv8o2M5TUNcnlMV6xQnSLAHWejE0dLL9+IJTdAFkRLHSTSafjvwVVdfcpknce6PslNJJTktztzXpf+uj0tx9MhfqDAyQ2EA6FO3w8FUAGCBDxucdsLZ1QHEiU1axwRFqrrRT+JWKkGq0H5HHQyhopySUy6VklJtpo=[/tex]