Let f : R → R be a continuous function and g : R → R be a Lipschitz function. Would you help me to prove that the system of differential equation x ′ = g ( x ) y ′ = f ( x ) ywith initial value x ( t 0 ) = x 0 and y ( t 0 ) = y 0 has a unique solution.Could I prove the uniqueness solution of x ′ = g ( x ), x ( t 0 ) = x 0 by Gronwall Inequality first then use the result to prove the second?

Question

Let   f  :  R  →  R be a continuous function and   g  :  R  →  R be a Lipschitz function. Would you help me to prove that the system of differential equation      x    ′    =  g  (  x  )      y    ′    =  f  (  x  )  ywith initial value   x  (      t    0    )  =      x    0   and   y  (      t    0    )  =      y    0   has a unique solution.Could I prove the uniqueness solution of       x    ′    =  g  (  x  ),   x  (      t    0    )  =      x    0   by Gronwall Inequality first then use the result to prove the second?

alisonhleel3 · Accepted Answer

Your reasonning is correct. Since   g is Lipschitz and the first equation of the system involves only   x, there is a unique solution   x  (  t  ) such that   x  (      t    0    )  =      x    0  .The second equation becomes      y    ′    =  f  (  x  (  t  )  )    y  ,    y  (      t    0    )  =      y    0    .It is a linear equation and has a unique solution, given by  y  (  t  )  =      y    0    e                        ∫                        t          0                t            f      (      x      (      s      )      )      d      s        .

Gabriella Sellers · Accepted Answer

My answer is suppose   (      x    1    (  t  )  ,      y    1    (  t  )  ) and   (      x    2    (  t  )  ,      y    2    (  t  )  ) is solution. First, write       x    1    (  t  )  =      x    0    +      ∫            t      0        t        x    ′    (  s  )  d  s the do the same for       x    2    (  t  ) and get   |      x    1    −      x    2    |  =  |      ∫            t      0        t    g  (      x    1    )  −  g  (      x    2    )  d  s  |  ≤  K      ∫            t      0        t    |      x    1    −      x    2    |  d  s. I doubt about this separation process.

Let f : <mi mathvariant="double-struck">R → <mi mathvariant="double

Answered question

Answer & Explanation

New Questions in Algebra I