What is the general solution of the differential equation d y d x - 2 x y = x ?

Name: What is the general solution of the differential equation d y d x - 2 x y = x ?
Uploaded: 2022-02-23T17:22:57+00:00
Description: What is the general solution of the differential equation d y d x - 2 x y = x ?

Question

What is the general solution of the differential equation                     d        y                    d        x              -    2    x    y    =    x  ?

brodireo1 · Accepted Answer

d        y                    d        x              -    2    x    y    =    x  Which we can write as:                    d        y                    d        x              =    2    x    y    +    x        ∴                  d        y                    d        x              =          (      2      y      +      1      )        x        ∴          1              2        y        +        1                            d        y                    d        x              =    x  Which is a first order separable differential equation, so we can &quot;separate the variables&quot; to get:      ∫               1              2        y        +        1                         d      y        =    ∫         x               d      x      Integrating we get, the General Solution:            1      2        ln          |      2      y      +      1      |        =          1      2              x      2        +    C  Applying the initial condition y(1)=1 we find:            1      2              ln      3        =          1      2        +    C    ⇒    C    =          1      2              ln      3        -          1      2      So we can write an implicit particular solution as:            1      2        ln          |      2      y      +      1      |        =          1      2              x      2        +          1      2              ln      3        -          1      2      We typically require an explicit solution, so we can rearrange as follows:      ln          |      2      y      +      1      |        =          x      2        +          ln      3        -    1        ∴          |      2      y      +      1      |        =          e                        x          2                +                  ln          3                -        1            Noting that the exponential function is positive over its entire domain, (as             e      x        &amp;gt;    0    ∀    x    ∈    ℝ  ):      2    y    +    1    =          e                        x          2                -        1                    e                        ln          3                          ∴    2    y    =    3          e                        x          2                -        1              -    1        ∴    y    =          3      2              e                        x          2                -        1              -          1      2

What is the general solution of the differential equation dy/dx−2xy=x?

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