Solving Differential Equations w/ polynomialsI'm new to solving differential equations.How would we go about solving a differential equation like y ′ = − 4 + 5 y − y 2 ?

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Solving Differential Equations w/ polynomialsI&#039;m new to solving differential equations.How would we go about solving a differential equation like      y    ′    =  −  4  +  5  y  −      y    2  ?

Adalyn Pitts · Accepted Answer

Step 1            d      y              d      t        =  −  4  +  5  y  −      y    2              d      y              −      4      +      5      y      −              y        2              =  d  t  ∫            d      y              4      +      5      y      −              y        2              =  ∫  d  tStep 2      1    3    ln  (      |              y      −      1              y      −      4            |    )  =  t  +      c    1    ln  (      |        3          y      −      4        +  1      |    )  =  3  t  +      c    2  From here just you can isolate for y, or just leave it implicit.

Bairaxx · Accepted Answer

Step 1One more way, use   y  =      u    ′        /    u, then      y    ′    =  −  4  +  5  y  −      y    2      ⟹              u              u        ″            −              u                  ′          2                            u      2        =  −  4  +  5            u      ′        u    −            u              ′        2                    u      2          ⟹        u    ″    −  5      u    ′    +  4  =  0Step 2Let   u  =      e          m      x      , then       m    2    −  5  m  +  2  =  0    ⟹    m  =  2  ,  3. So   u  (  x  )  =      C    1        e          2      x        +      C    2        e          3      x      , which gives  y  (  x  )  =            2              C        1                    e                  2          x                    +      3              C        2                    e                  3          x                                    C        1                    e                  2          x                    +              C        2                    e                  3          x                      =            2      +      3      C              e                  x                            1      +      C              e                  x

Solving Differential Equations w/ polynomials. I'm new to solving differential equations. How would we go about solving a differential equation like y'=-4+5y-y^2?

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