I've found this particular equation rather tough, can you give me some hints on how to solve y ˙ + t cos ⁡ π y 2 + 1 − t = y

Question

I&#039;ve found this particular equation rather tough, can you give me some hints on how to solve            y      ˙        +  t  cos  ⁡            π      y        2    +  1  −  t  =  y

Zachery Conway · Accepted Answer

y      ˙        +  t  cos  ⁡                    π        y            2        +  1  −  t  =  y                    d        y                    d        t              =      (    1    −    cos    ⁡                            π          y                2              )    t  +  y  −  1            y      ˙        +  t  cos  ⁡                    π        y            2        +  1  −  t  =  yLet   u  =  y  −  1,Then   y  =  u  +  1                    d        y                    d        t              =                    d        u                    d        t              ∴                    d        u                    d        t              =            (        1  −  cos  ⁡                    π        (        u        +        1        )            2                  )        t  +  u      (          (      1      −      cos      ⁡              (                                            π              u                        2                          +                              π            2                          )            )        t    +    u    )                      d        t                    d        u              =  1      (          (      1      +      sin      ⁡                                    π            u                    2                    )        t    +    u    )                      d        t                    d        u              =  1Let   v  =  t  +            u              1        +        sin        ⁡                                            π              u                        2                              ,Then   t  =  v  −            u              1        +        sin        ⁡                                            π              u                        2                                                  d        t                    d        u              =                    d        v                    d        u              +                                                        π              u                        2                          cos        ⁡                                            π              u                        2                                                (          1          +          sin          ⁡                                                    π                u                            2                                )                2              −            1              1        +        sin        ⁡                                            π              u                        2                                ∴      (    1    +    sin    ⁡                            π          u                2              )    v      (                            d          v                          d          u                      +                                                                      π                u                            2                                cos          ⁡                                                    π                u                            2                                                            (            1            +            sin            ⁡                                                            π                  u                                2                                      )                    2                      −                  1                  1          +          sin          ⁡                                                    π                u                            2                                            )    =  1      (    1    +    sin    ⁡                            π          u                2              )    v                    d        v                    d        u              +      (                                                                      π                u                            2                                cos          ⁡                                                    π                u                            2                                                1          +          sin          ⁡                                                    π                u                            2                                            −    1    )    v  =  1      (    1    +    sin    ⁡                            π          u                2              )    v                    d        v                    d        u              =      (    1    −                                                                      π                u                            2                                cos          ⁡                                                    π                u                            2                                                1          +          sin          ⁡                                                    π                u                            2                                            )    v  +  1  v                    d        v                    d        u              =      (                  1                  1          +          sin          ⁡                                                    π                u                            2                                            −                                                                      π                u                            2                                cos          ⁡                                                    π                u                            2                                                            (            1            +            sin            ⁡                                                            π                  u                                2                                      )                    2                      )    v  +            1              1        +        sin        ⁡                                            π              u                        2                              This belongs to an Abel equation of the second kind.In fact all Abel equation of the second kind can be transformed into Abel equation of the first kind.Let   v  =            1      w      ,Then                     d        v                    d        u              =  −            1              w        2                                d        w                    d        u              ∴  −            1              w        3                                d        w                    d        u              =      (                  1                  1          +          sin          ⁡                                                    π                u                            2                                            −                                                                      π                u                            2                                cos          ⁡                                                    π                u                            2                                                            (            1            +            sin            ⁡                                                            π                  u                                2                                      )                    2                      )              1      w        +            1              1        +        sin        ⁡                                            π              u                        2                                                  d        w                    d        u              =  −                    w        3                    1        +        sin        ⁡                                            π              u                        2                                +      (                                                                      π                u                            2                                cos          ⁡                                                    π                u                            2                                                            (            1            +            sin            ⁡                                                            π                  u                                2                                      )                    2                      −                  1                  1          +          sin          ⁡                                                    π                u                            2                                            )        w    2

I've found this particular equation rather tough, can you give me some hints on how to solve <

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