I was given a problem, it has been worded as follows: Use the substitution z = y x to transform the differential equation d y d x = − 3 x y y 2 − 3 x 2 , into a linear equation. Hence obtain the general solution of the original equation.Workings: d y d x = − 3 ( y x ) ( y x ) 2 − 3 Multiplied by 1 x 2 1 x 2 d y d x = − 3 z z 2 − 3 z ′ = y ′ x − y x 2 ( z ′ + y x 2 ) x = − 3 z z 2 − 3 After a whole page of workings, I arrive at this z ′ + z x = 3 z Technically, this is a linear equation since it takes the form y'+p(x)y=f(x) . Since z=f(y,x). However, I'm unable to compute this, since I won't know how to integrate z with respect to x. I do know this can easily be done by separating variables, however, we are basically told to solve it this way.

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I was given a problem, it has been worded as follows: Use the substitution   z  =      y    x   to transform the differential equation                     d        y                    d        x              =                  −        3        x        y                              y          2                −        3                  x          2                    , into a linear equation. Hence obtain the general solution of the original equation.Workings:                    d        y                    d        x              =                  −        3        (                  y          x                )                    (                  y          x                          )          2                −        3            Multiplied by             1              x        2                    1              x        2                        d      y              d      x        =            −      3      z                      z        2            −      3            z    ′    =            y      ′        x    −      y          x      2        (      z    ′    +      y          x      2        )  x  =            −      3      z                      z        2            −      3      After a whole page of workings, I arrive at this      z    ′    +      z    x    =      3    z  Technically, this is a linear equation since it takes the form y&#039;+p(x)y=f(x) . Since z=f(y,x). However, I&#039;m unable to compute this, since I won&#039;t know how to integrate z with respect to x. I do know this can easily be done by separating variables, however, we are basically told to solve it this way.

bamenyab4mxn · Accepted Answer

From y=zx, draw      y    ′    =      z    ′    x  +  z  .Then      z    ′    x  +  z  =  −            3      z                      z        2            −      3      which is separable:      z    ′    x  =  −  z  −            3      z                      z        2            −      3        =  −            z      3                      z        2            −      3        .The equation can be seen as linear (and homogeneous) by swapping the roles of the dependent and independent variables and rewriting            d      x              d      z        +                    z        2            −      3              z      3        x  =  0.This doesn&#039;t ease the solution.

llunallenaipg5r · Accepted Answer

Setting  y  =  u  xthen we have      u    ′    x  +  u  =  −            3      u                      u        2            −      3      simplifying we get      u    ′    x  =  −  u      (          3                        u          2                −        3              +    1    )  or      u    ′    x  =  −            u      3                      u        2            −      3      this equation is separble and it follows  −                    u        2            −      3              u      3        d  u  =            d      x        x

I was given a problem, it has been worded as follows: Use the substitution z = y

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