Differential equation for a spring mass system m y ″ + β y ′ + k y = 0I have to find the displacement y at any time t. β = 2 m k and the initial conditions are y ( 0 ) = h &gt; 0 and y ′ ( 0 ) = v. I initally found the following y = ( a + b t ) e − t m k m I am currently stuck with the initial conditions as I end up with b = − v m k a k from y ′ ( 0 ) = v and I don't know how to interpret y ( 0 ) = h &gt; 0Now have y = ( h + v + h t k m ) e − t m k m and have to express y=0 as an inequality v &lt; − f ( h , k , m ) for the function f

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Differential equation for a spring mass system  m      y    ″    +  β      y    ′    +  k  y  =  0I  have to find the displacement y at any time   t.   β  =  2      m    k   and the initial conditions are   y  (  0  )  =  h  &amp;gt;  0 and       y    ′    (  0  )  =  v. I initally found the following  y  =  (  a  +  b  t  )      e                  −        t                  m          k                    m      I am currently stuck with the initial conditions as I end up with   b  =            −      v              m        k                    a      k       from      y    ′    (  0  )  =  v and I don&#039;t know how to interpret   y  (  0  )  =  h  &amp;gt;  0Now have  y  =  (  h  +  v  +  h  t            k      m        )      e                  −        t                  m          k                    m      and have to express y=0 as an inequality   v  &amp;lt;  −  f  (  h  ,  k  ,  m  ) for the function   f

Zayden Wiley · Accepted Answer

y  =      (    h    +    t    (    v    +    h                            k          m                      )    )        e                  −        t                  m          k                    m        =  0then   h  +  t  (  v  +  h                    k        m              )  =  0 or  t  =  −            h              v        +        h                                            k              m                                            &amp;gt;  0with   h  &amp;gt;  0 sould be   v  +  h                    k        m              &amp;lt;  0 or   v  &amp;lt;  −  h                    k        m              =  −  f  (  h  ,  k  ,  m  )

Differential equation for a spring mass system m y ″ </msup> + β<

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