Find the solution of the given initial value problem. y ′ + ( 2 / t ) y = ( cos ⁡ t ) t 2 , y ( 3.14 ) = 0 , t &gt; 0

Question

Find the solution of the given initial value problem.      y    ′    +  (  2      /    t  )  y  =  (  cos  ⁡  t  )      t          2        ,  y  (  3.14  )  =  0  ,  t  &amp;gt;  0

repotasonwf · Accepted Answer

Given equation is      y          1        +      2    t    y  =            cos      ⁡      t              t              2              →  (  1  )This is a linear equation of first order in ywhere   P  (  t  )  =      2    t    a  n  d  Q  (  t  )  =            cos      ⁡      t              t              2              ∫  P  (  t  )  d  t  =  ∫      2    t    d  t  =  2  log  ⁡  t  =  log  ⁡      t          2      Integrating Factor   =      e          ∫      P      (      t      )      d      t        =      e          log      ⁡              t                  2                      =      t          2      Multiplying (1) with       t          2      , we get      t          2            y          1        +  2  t  y  =  cos  ⁡  t  →  (  2  )(2) can be written as      d          d      t        [  y      t          2        ]  =  cos  ⁡  t  ∫      d          d      t        [  y      t          2        ]  d  t  =  ∫  cos  ⁡  t  d  t  ⇒  y  ∗      t          2        =  sin  ⁡  t  +  C  ⇒  y  (  t  )  =            sin      ⁡      t              t              2              +      C          t              2            we know that   y  (  π  )  =  0so             sin      ⁡      π              π              2              +      C          π              2              =  0  ⇒  C  =  0Therefore the general solution of (1) is   y  =            sin      ⁡      t              t              2

Find the solution of the given initial value problem. y'+(2/t)y=(cos t)t^2 , y(3.14)=0, t>0

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