Solve the following initial value problem: ( 1 − t 2 ) y t − 4 t y = 1 , y ( 2 ) = − 1What is the largest interval on which its solution is guaranteed to uniquely exist?

Question

Solve the following initial value problem:  (  1  −      t          2        )      y          t        −  4  t  y  =  1  ,  y  (  2  )  =  −  1What is the largest interval on which its solution is guaranteed to uniquely exist?

neobuzdanio · Accepted Answer

The homogeneous differential equation can be solved by separation of variables.  d  y      /    y  =  4  t  d  t      /    (  1  −      t          2        ) is the separated differential equation.Integration   ln  ⁡  (  y  )  =  −  2  ln  ⁡  (  −  1  +      t          2        ). Exponentiation:   y  (  t  )  =  a  (      t          2        −  1      )          −      2        .  y  (  2  )  =  a      /    9  =  −  1.  a  =  −  9.  y  (  t  )  =  −  9      /    (      t          2        −  1      )          2      .The inhomogeneous differential equation is solved by:  y  (  t  )  =  (  −  25  +  3  t  −      t          2        )      /    (  3  (  −  1  +      t          2            )          2        ).This is derived by variation of constants.  y  (  t  )  =  f  (  t  )  (  −  1  +      t          2            )          −      2        .      y    ′    (  t  )  =      f    ′    (  t  )  (  −  1  +      t          2            )          2        −  4  t  f  (  t  )  (  −  1  +      t          2            )          −      3      .So in the inhomogeneous differential equation:   −      f    ′    (  t  )  (  −  1  +      t          2            )          −      1        +  4  t  f  (  t  )  (  −  1  +      t          2            )          −      2        −  4  t  f  (  t  )  (  −  1  +      t          2            )          −      2        =      f    ′    (  t  )  (  −  1  +      t          2            )          −      1        =  1. Simply integrate and adopt the constant to rewrite the above result.

Solve the following initial value problem: (1-t^2)y'-4ty =1, y(2)=-1 What is the largest interval on which its solution is guaranteed to uniquely exist?

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