How to solve the system d x d t t = − x + y t d y d t t = − 2 x + y t

Question

How to solve the system            d      x              d      t        t  =  −  x  +  y  t            d      y              d      t        t  =  −  2  x  +  y  t

Donavan Scott · Accepted Answer

(  t  D  +  1  )  x  =  t  y and   (  t  D  −  t  )  y  =  −  2  x thus       1          −      2        (  t  D  +  1  )  (  t  D  −  t  )  y  =  t  y. Now substitute   y  =      t    r   and find a condition for   r then you can derive   x. This is not a Cauchy Euler system.Added After Original Answer overlooked an unfortunate   t: If we write   t            d      x              d      t        +  x  =  y  t and   t            d      y              d      t        +  2  x  =  y  t then subtracting yields:  t      d          d      t            [    y    −    x    ]    +  x  =  0Let   w  =  y  −  x hence   y  =  w  +  x and we find:  t            d      w              d      t        +  x  =  0    &amp;amp;    t            d      x              d      t        +  x  =  t  (  w  +  x  )Eliminating   x via   x  =  −  t            d      w              d      t       yields:  t                    d        2            w              d              t        2              +  (  2  −  t  )            d      w              d      t        +  w  =  0.I think we can solve this by the series method, then   x  =  −  t            d      w              d      t       hence we can calculate that and find   y from   y  =  w  +  x.

How to solve the system d x </mrow> d t </m

Answered question

Answer & Explanation

New Questions in Algebra I