Fourth solution of y ′ = 10 3 x y 2 / 5 , y ( 0 ) = 0?By inspection, we know that y = 0 is a solution. If we separate variables, we get another solution, y = x 10 / 3 . The book tells me this much, and asks me to find the other 2 solutions that differ on every open interval containing x = 0 and are defined on ( − ∞ , ∞ ).I found the third solution (by guessing) to be y = − x 10 / 3 . The only other possibility i could think about is x = 0, but I'm pretty sure that's not valid since y′ would be undefined. What is the last solution? How do we know there are only 4 possible solutions (and not more)?

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Fourth solution of       y    ′    =            10      3        x      y          2              /            5      ,   y  (  0  )  =  0?By inspection, we know that   y  =  0 is a solution. If we separate variables, we get another solution,   y  =      x          10              /            3      . The book tells me this much, and asks me to find the other 2 solutions that differ on every open interval containing   x  =  0 and are defined on   (  −  ∞  ,  ∞  ).I found the third solution (by guessing) to be   y  =  −      x          10              /            3      . The only other possibility i could think about is   x  =  0, but I&#039;m pretty sure that&#039;s not valid since y′ would be undefined. What is the last solution? How do we know there are only 4 possible solutions (and not more)?

Abagail Meyers · Accepted Answer

Step 1  y  =  −      x          10              /            3       is not a solution. It is decreasing where it should be increasing and increasing where it should be decreasing.You could have any one of   y  =  0 and   y  =      x          10              /            3       for   x  ≥  0, and any one of those for   x  &amp;lt;  0. That makes 4 different solutions.Step 2There are also solutions of the form   y  =      {                            (                      x            2                    −                      a            2                                )                          5                              /                            3                                                             for                           x              &amp;lt;              a                                                            0                                       for                           a              ≤              x              ≤              b                                                            (                      x            2                    −                      b            2                                )                          5                              /                            3                                                             for                           x              &amp;gt;              b                                               where   −  ∞  ≤  a  &amp;lt;  0  &amp;lt;  b  ≤  +  ∞, but these don&#039;t differ on every open interval containing 0.

Fourth solution of y'=10/3 xy^{2/5}, y(0)=0?

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