Find the solution to the differential equationAssume x &gt; 0 and let x ( x + 1 ) d u d x = u 2 , u ( 1 ) = 4.I started off by doing some algebra to get: 1 u 2 d u = 1 x 2 + x d x .I then took the partial fraction of the right side of the equation: 1 u 2 d u = ( 1 x − 1 x + 1 ) .I then took the integral of both sides: − 1 u = log ⁡ x − log ⁡ ( x + 1 ) + C .From here I don't know what to do because we are solving for u ( x ) and I'm not sure how to get that from − 1 u

Question

Find the solution to the differential equationAssume   x  &amp;gt;  0 and let  x  (  x  +  1  )            d      u              d      x        =      u    2    ,  u  (  1  )  =  4.I started off by doing some algebra to get:      1          u      2        d  u  =      1                  x        2            +      x        d  x  .I then took the partial fraction of the right side of the equation:      1          u      2        d  u  =      (          1      x        −          1              x        +        1              )    .I then took the integral of both sides:  −      1    u    =  log  ⁡      x    −  log  ⁡      (    x    +    1    )    +  C  .From here I don&#039;t know what to do because we are solving for   u  (  x  ) and I&#039;m not sure how to get that from   −      1    u

grizintimbp · Accepted Answer

Solving for             d      u              d      x       we have            d      u              d      x        =            u      2              x      (      x      +      1      )          ⇒            d      u              d      x        =            u      2                      x        2            +      x                            d        u                    d        x                    u      2        =      1                  x        2            +      x        .Integrate both sides &amp;amp; evaluate the integrals:  −      1    u    =  log  ⁡  (  x  )  −  log  ⁡  (  x  +  1  )  +      C    1      ⇒  u  =  −      1          log      ⁡      (      x      )      −      log      ⁡      (      x      +      1      )      +              C        1              .Now apply the initial condition:  −      1                  C        1            −      log      ⁡      (      2      )        =  4  ⇒      C    1    =      1    4    (  −  1  +  4  log  ⁡  (  2  )  )  .This gives the result.