Consider the differential system { y ′ ( t ) = a y ( t ) 3 + b z ( t ) z ′ ( t ) = c z ( t ) 5 − b y ( t ) with t &gt; 0 y ( 0 ) = y 0 , z ( 0 ) = z 0 , a &lt; 0 , c &lt; 0 , b ∈ Rthe question is to prouve that this system admits a unique solution on [ 0 , + ∞ ]?

Question

Consider the differential system      {                                                y            ′                    (          t          )          =          a          y          (          t                      )            3                    +          b          z          (          t          )                                                          z            ′                    (          t          )          =          c          z          (          t                      )            5                    −          b          y          (          t          )                        with   t  &amp;gt;  0  y  (  0  )  =      y    0    ,  z  (  0  )  =      z    0    ,    a  &amp;lt;  0  ,    c  &amp;lt;  0  ,    b  ∈  Rthe question is to prouve that this system admits a unique solution on   [  0  ,  +  ∞  ]?

Erick Blake · Accepted Answer

By standard ODE theorems, we know that if a solution doesn&#039;t exist, it will only not exist because the solution &quot;blows up,&quot; that is, if there exists a   T  &amp;gt;  0 such that   x  (  t  ) or   y  (  t  ) is unbounded as  t  →      T    −  .Multiply the first equation by   y  (  t  ), and the second solution by  z  (  t  ), and then add them together, you obtain      1    2        d          d      t        (  (  y  (  t  )      )    2    +  (  z  (  t  )      )    2    )  =  a  (  y  (  t  )      )    4    +  c  (  z  (  t  )      )    6    .Since   a and   c are negative, it follows that   (  y  (  t  )      )    2    +  (  z  (  t  )      )    2   is bounded. Hence a blow up can never happen.

Consider the differential system { <mtable columnalign="left left" rowspacing=".2em" co

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