I got the following system of non-linear equations that I need to solve for v e , v s and ω: m p v a = m s v s + m p v e l 2 m p v a = l 2 m p v e + I ω 1 2 m p v a 2 = 1 2 m p v e 2 + 1 2 m s v s 2 + 1 2 I ω 2 I know how to solve this with substitution and a lot of scratch paper.With linear systems, one can just derive the matrix and use gauss-jordan / reduced row echolon form and then the solution is directly apparent.Is there something handy for non-linear equations as well?

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I got the following system of non-linear equations that I need to solve for       v    e  ,       v    s   and   ω:      m    p        v    a    =      m    s        v    s    +      m    p        v    e        l    2        m    p        v    a    =      l    2        m    p        v    e    +  I  ω      1    2        m    p        v    a    2    =      1    2        m    p        v    e    2    +      1    2        m    s        v    s    2    +      1    2    I      ω    2  I know how to solve this with substitution and a lot of scratch paper.With linear systems, one can just derive the matrix and use gauss-jordan / reduced row echolon form and then the solution is directly apparent.Is there something handy for non-linear equations as well?

Raina Russo · Accepted Answer

For your example, you have a &quot;nearly&quot; linear system, because only one equation is quadratic. In that case, you can express all solutions of the linear part of the system in the form       x    0    +  α      x    h  , substitute that expression into the quadratic equation and solve the resulting equation for   α.If you apply this technique to your system, you get       v    e    =      v    a    −  α,       v    s    =            m      p              m      s        α and   ω  =            l              m        p                    2      I        α. If you substitute this into the quadratic equation      m    p    (      v    a    2    −      v    e    2    )  =      m    s        v    s    2    +  I      ω    2  you get      m    p    (  2      v    a    −  α  )  α  =            m      p      2              m      s            α    2    +                    l        2                    m        p        2                    4      I            α    2  One solution of this quadratic equation is obviously   α  =  0, but I guess you are more interested in the other solution. Assuming       m    p    α  ≠  0, we can divide by       m    p    α to get   2      v    a    −  α  =            m      p              m      s        α  +                    l        2                    m        p                    4      I        α. It&#039;s easy to solve this equation for   α.

I need to solve for v_e, v_s and ω: m_pv_a=m_sv_s+m_pv_e l/2m_pv_a=l/2m_pv_e+Iω 1/2m_pv^2_a=1/2m_pv^2_e+1/2m_sv^2_s+1/2Iω^2

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