Solution of a system of differential equations x ′ = − y + x y 2 y ′ = 4 x − 4 x 2 y x ′ = − y + x y 2 y ′ = 4 x − 4 x 2 yproposed solution to this which uses the fact that d y d x = y ′ x ′ = 4 x ( 1 − x y ) y ( x y − 1 ) = − 4 x y and then solves this as a separable equation, which is straight forward enough. What I am struggling to understand is why one chooses to calculate d y d x . It seems to me quite arbitrary to calculate this derivative. What is the connection between this derivative and the original system?

Question

Solution of a system of differential equations      x    ′    =  −  y  +  x      y    2          y    ′    =  4  x  −  4      x    2    y      x    ′    =  −  y  +  x      y    2          y    ′    =  4  x  −  4      x    2    yproposed solution to this which uses the fact that            d      y              d      x        =            y      ′              x      ′        =            4      x      (      1      −      x      y      )              y      (      x      y      −      1      )        =  −            4      x        y  and then solves this as a separable equation, which is straight forward enough. What I am struggling to understand is why one chooses to calculate             d      y              d      x      . It seems to me quite arbitrary to calculate this derivative. What is the connection between this derivative and the original system?

livin4him777lf · Accepted Answer

Consider system      x    ˙    =  f  (  x  ,  y  )  ,        y    ˙    =  g  (  x  ,  y  )  ,where           ˙   denotes derivative with respect to   t. Its solutions are given by   (  x  (  t  )  ,  y  (  t  )  ) and can be represented as curves in 3-dimensional space   R  ×        R    2  . However, the system is autonomous, which means that   f and   п do not depend on   t explicitly. In this case instead of dealing with curves in 3D space, we can stick to the plane   (  x  ,  y  ), in which the solution curves turns into an orbit   (  x  (  t  )  ,  y  (  t  )  ) where   t is now a parameter. Hence an orbit is a projection of the solution curve on the plane   (  x  ,  y  ) with a specific time direction (we can put arrows on our orbits).Finally, how to find actually the orbits. We can solve the original system and then eliminate   t. Or, differently, we can eliminate   t from the original system and consider, using the fact that

Solution of a system of differential equations x ′ </msup> = −<!--

Answered question

Answer & Explanation

New Questions in Algebra I