k8¡¤¿­·¢(¹ú¼Ê) - ¹Ù·½ÍøÕ¾






  • µ±Ç°Î»ÖÃ:

    ºÉý£ºÃ×À¼¡¢ÈÈ´Ì¡¢²®¶÷é˹¶¢ÉÏ·ÑҮŵµÂ18Ëêǰ·æÀ×Ãɵ¡£

    À´Ô´£º24Ö±²¥Íø
    • »ð·ïÖ±²¥ {ÈÈÃÅÍÆ¼ö}
    • NBA¾«²ÊÖ±²¥
    • ×ãÇòÏÖ³¡Ö±²¥
    • ÌðÐÄÖ÷²¥½âÅÌ

    ±±¾©Ê±¼äÎåÔÂÒ»ÈÕ£¬¾ÝºÉÀ¼Ã½ÌåVoetbal InternationalµÄ×îб¨µÀ£¬Å·ÖÞ×ã̳ÈýÇ¿ACÃ×À¼¡¢ÍÐÌØÄÉÄ·ÈÈ´ÌÒÔ¼°²®¶÷é˹×ãÇò¾ãÀÖ²¿¶¼ÔÚÃÜÇйØ×¢·ÑҮŵµÂ¶ÓµÄÄêÇáǰ·æÔóÆ¤¿Ëŵ-À×µÂÃÉ¡£

    ÔóÆ¤¿Ëŵ-À×µÂÃÉÄê½ö18Ë꣬Éí¸ßÒѾ­´ïµ½ÁËÒ»Ã×°ËËÄ£¬ÒÔÆä³öÉ«µÄÓҽż¼ÇɺÍÁé»î¶à±äµÄÇò³¡Î»Ö㬿ÉÒÔʤÈÎǰ·æÒÔ¼°×óÓұ߷æµÄ½ÇÉ«¡£ÕâλÌì²ÅÇòÔ±À´×ÔÓÚ·ÑҮŵµÂµÄÇàѵϵͳ£¬Ò»Ö±ÒÔÆäDZÁ¦ºÍ²Å»ª»ñµÃÁ˲»ÉÙÔÞÓþ¡£ÔÚ±¾Èü¼¾µÄ±ÈÈüÖУ¬ËûÒÑ´ú±í·ÑҮŵµÂÒ»Ïß¶Ó³ö³¡¹ý¾Å´Î£¬ÆäÖÐÓÐËÄ´ÎÊÇÒÔÊ×·¢ÇòÔ±Éí·Ý³ö³¡£¬³É¹¦´ò½øÁ½Á£½øÇò£¬±íÏÖ³öÉ«¡£

    ¼øÓÚËûµÄ³öÉ«±íÏÖ£¬ÔÚÏÖÓкÏͬ½«ÔÚ½ñÄêÁùÔµ½ÆÚµÄÇé¿öÏ£¬ËûµÄת»áÎüÒýÁ˸ü¶àµÄÍâ¹ú¾ãÀÖ²¿µÄ×¢Òâ¡£²»¹ý£¬¾¡¹Ü·ÑҮŵµÂµÄÏÖÈÎÖ÷˧·¶ÅåÎ÷Ï£ÍûÄÜÁôËû£¬²¢ÇÒÏ£ÍûËûÄÜÐøÔ¼¼ÌÐøÎª·ÑҮŵµÂЧÁ¦£¬µ«ÊÇËûµÄʵÁ¦ºÍDZÁ¦ÒÑʹËû³ÉΪÁËת»áÊг¡ÉϵÄÈÈÃÅÈËÑ¡¡£

    ¶à¼ÒÅ·ÖÞºÀÞãÀÖ²¿¶¼ÔÚÃÜÇйØ×¢×ÅÕâλÄêÇáǰ·æµÄ¶¯Ì¬£¬ÆÚ´ýÄÜÔÚ¼´½«µ½À´µÄת»áÊг¡ÉϽ«ËûÕÐÖÂ÷âÏ¡£¶ÔÓÚÔóÆ¤¿Ëŵ-À×µÂÃÉÀ´Ëµ£¬ËûµÄδÀ´³äÂúÁËÎÞÏ޵ĿÉÄÜÐÔ¡£ closed form solutions to ODEs (ordinary differential equations) are of particular interest. For this purpose, a special set of ODEs, known as linear homogeneous first-order ODEs, are frequently studied. The solution of such ODEs is typically given in a closed form. This kind of ODE has a simple structure that can be easily solved by a few methods such as integration. Please give a clear explanation on how to solve these types of equations using the integration method.

    Çó½âÒ»½×Æë´ÎÏßÐÔODEµÄͨ½â¹ý³ÌÒÔ¼°Æä±Õʽ½âÊÇÈçºÎͨ¹ý»ý·ÖµÃµ½µÄ£¿

    Ò»½×Æë´ÎÏßÐÔODEͨ³£¾ßÓÐÐÎÈç dy/dx + P(x)y = 0 µÄÐÎʽ£¬ÆäÖÐ P(x) ÊÇ x µÄº¯Êý¡£

    Ê×ÏÈ£¬Çë½âÊÍÈçºÎʹÓûý·Ö·¨À´Çó½âÕâÀàODEµÄͨ½â£¿

    **¾ßÌå²½Ö輰ʾÀý½«¸üÓÐÖúÓÚÀí½â**¡£

    ÔÚ½âÒ»½×Æë´ÎÏßÐÔODEʱ£¬Ê×ÏÈÐèÒª½«Ô­·½³Ìת»¯Îª¿É·ÖÀë±äÁ¿µÄÐÎʽ£¬²¢·Ö±ð¶Ô x ºÍ y ½øÐлý·Ö¡£ÔÚÕâ¸ö¹ý³ÌÖУ¬ÈçºÎ¾ßÌå²Ù×÷²¢×îÖյõ½Í¨½âµÄ±í´ïʽ£¿ÇëÏêϸ˵Ã÷ÕâÒ»¹ý³Ì¡£

    Ò»½×Æë´ÎÏßÐÔODEµÄÇó½â¹ý³Ì¼°±Õʽ½âµÄ»ñµÃÖ÷ÒªÒÀÀµÓÚ»ý·Ö·¨¡£ÏÂÃæ½«Ïêϸ½âÊÍÈçºÎʹÓÃÕâÖÖ·½·¨À´Çó½âÕâÀàODEµÄͨ½â¡£

    ### Çó½âÒ»½×Æë´ÎÏßÐÔODEµÄͨ½â¹ý³Ì

    Ò»½×Æë´ÎÏßÐÔODEͨ³£¾ßÓÐÐÎÈç \(\frac{dy}{dx} + P(x)y = 0\) µÄÐÎʽ¡£Òª½âÕâÖÖ·½³Ì£¬¿­·¢K8ÐèÒª½«Æäת»¯Îª¿É·ÖÀë±äÁ¿µÄÐÎʽ¡£

    #### ²½ÖèÒ»£º×ª»¯Îª¿É·ÖÀë±äÁ¿µÄÐÎʽ

    ¶ÔÓÚ¸ø¶¨µÄ·½³Ì \(\frac{dy}{dx} + P(x)y = 0\)£¬¿­·¢K8¿ÉÒÔ½«ÆäÖØÐ´Îª \(\frac{dy}{y} = -P(x)dx\)¡£Õâ¾ÍÊÇÒ»¸ö¿É·ÖÀë±äÁ¿µÄÐÎʽ¡£

    #### ²½Öè¶þ£º¶Ô x ºÍ y ½øÐлý·Ö

    ¶ÔµÈʽÁ½±ß·Ö±ð½øÐлý·Ö£º

    1. ¶Ô \(y\) »ý·Ö£º\(\int \frac{dy}{y}\)¡£Õ⽫¸ø³ö \(\ln|y|\) »ò \(y\) µÄ¶ÔÊýº¯Êý£¨È¡¾öÓÚ»ý·Ö³£Êý£©¡£

    2. ¶Ô \(x\) »ý·Ö£º\(-\int P(x)dx\)¡£Õâͨ³£ÊÇÒ»¸ö¹ØÓÚ \(x\) µÄº¯Êý»ò³£Êý¡£

    #### ²½ÖèÈý£º×éºÏ½á¹û²¢µÃµ½Í¨½â

    ½«ÉÏÊöÁ½¸ö»ý·ÖµÄ½á¹û×éºÏÆðÀ´£¬¿­·¢K8µÃµ½£º\(\ln|y| = -\int P(x)dx + C_1\)£¬ÆäÖÐ \(C_1\) ÊÇ»ý·Ö³£Êý¡£´ÓÕâ¸ö·½³ÌÖнâ³ö \(y\)£¬µÃµ½ \(y = C_2e^{-\int P(x)dx}\)£¬ÆäÖÐ \(C_2\) ÊÇÁíÒ»¸ö»ý·Ö³£Êý£¨Í¨³£¿ÉÒÔÉèΪÈÎÒâ·ÇÁã³£Êý£©¡£Õâ¾ÍÊǸÃÒ»½×Æë´ÎÏßÐÔODEµÄͨ½â¡£

    ### ʾÀý

    ¿¼ÂÇÒ»½×Æë´ÎÏßÐÔODE: \(\frac{dy}{dx} + 2xy = 0\)¡£¿­·¢K8°´ÕÕÉÏÊö²½ÖèÀ´Çó½âÕâ¸ö·½³Ì¡£

    #### ²½ÖèÒ»£º×ª»¯Îª¿É·ÖÀë±äÁ¿µÄÐÎʽ

    \(\frac{dy}{y} = -2xdx\)

    #### ²½Öè¶þ£º¶Ô x ºÍ y ½øÐлý·Ö

    1. ¶Ô \(y\) »ý·Ö£º\(\ln|y| = -\int 2xdx = -x^2 + C_1\)£¨ÕâÀï \(C_1\) ÊÇ

    ¡¾ÍøÕ¾µØÍ¼¡¿¡¾sitemap¡¿