I need help with Differential equations.What is the Implicit answer for this differential equation? d y d x = y 2 − 4

Question

I need help with Differential equations.What is the Implicit answer for this differential equation?            d      y              d      x        =      y          2        −  4

Leon Webster · Accepted Answer

Step 1Here is how you solve equations like yours. You are trying to solve             d      y              d      x        =  f  (  y  ). Rewrite it as             d      y              f      (      y      )        =  d  x, integrate on both sides   ∫      1          f      (      y      )        d  y  =  ∫  d  x  =  x  +  c and invert to get expression for y(x). If you have conditions on the value of y(x) as some particular x, like   x  =  0, then you can determine c as well.Step 2Applying this to your example. You have   ∫      1                  y        2            −      4        d  y  =  x  +  c. The integral is       1    4    log  ⁡            (                        2          −          y                          2          +          y                    )       so that             2      −      y              2      +      y        =  exp  ⁡      (    4    x    +    4    c    )  . From here solving for y is straightforward.

Modelfino0g · Accepted Answer

Step 1            d      y              d      x        =      y    2    −  4    ⟺              d      y                      y        2            −      4        =  d  xThen integrate each side. You can easily use partial fractions on the left hand side.:  ∫            d      y                      y        2            −      4        =  ∫            d      y              (      y      −      2      )      (      y      +      2      )        =  ∫      A          y      −      2        +      B          y      +      2        A  y  +  2  A  +  B  y  −  2  B  =  1    ⟺    (  A  +  B  )  y  +  2  (  A  −  B  )  =  1    ⟺    (  A  +  B  =  0  ,    2  (  A  −  B  )  =  1  )    ⟺    (  A  =  −  B  ∧  −  4  B  =  1  )  .So   A  =      1    4    ,  B  =  −      1    4  That gives us      1    2    ∫      1          y      −      2        −      1          y      +      2        d  y  =  ∫  d  x  +  c                    (1)                            ⟺                          1          4                ln        ⁡                  (                                    y              −              2                                      y              +              2                                )                =        x        +        C            Step 2Given your last comment, we can continue as follows from (1):                    (2)                    ln        ⁡                  (                                    y              −              2                                      y              +              2                                )                =        4        x        +                  C          ∗                    Raising each side as a power of e gives us:  ln  ⁡      (                  y        −        2                    y        +        2              )    =  4  x  +  C    ⟺        e                  ln        ⁡                  (                                    y              −              2                                      y              +              2                                )                      =      e                  4        x        +        C                ⟺              y      −      2              y      +      2        =      e          4      x                                    e          C                ⏟                            C        ′            So                     (3)                                          y            −            2                                y            +            2                          =                  C          ′                          e                      4            x

What is the Implicit answer for this differential equation? (dy)/(dx)=y^2-4

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