Help with coupled differential equationsI have these two coupled differential equations a x ¨ + b x ⃛ + c tan ⁡ ( x + y ) x ˙ = 0 , a y ¨ − b y ⃛ − c tan ⁡ ( x + y ) y ˙ = 0with a , b , c = c o n s t .Now I basically have no experience at all with more advanced differential equations. Are these even solvable? I guess answering this type of question is already a science to itself, but maybe you can share some intuition under which circumstances one can hope to find a solution? What are possible methods or tricks?

Question

Help with coupled differential equationsI have these two coupled differential equations  a            x      ¨        +  b            x      ⃛        +  c  tan  ⁡  (  x  +  y  )            x      ˙        =  0  ,  a            y      ¨        −  b            y      ⃛        −  c  tan  ⁡  (  x  +  y  )            y      ˙        =  0with   a  ,  b  ,  c  =  c  o  n  s  t  .Now I basically have no experience at all with more advanced differential equations. Are these even solvable? I guess answering this type of question is already a science to itself, but maybe you can share some intuition under which circumstances one can hope to find a solution? What are possible methods or tricks?

Iademarco11 · Accepted Answer

Step 1The only exact solution I can find is  x  =      c    1    ,    y  =      c    2    ,    such that       c    1    +      c    2    ≠            (      2      n      +      1      )      π        2      ∀  n  ∈      Z    .You need the condition on the sum to prevent evaluating the tangent function at one of its vertical asymptotes.Step 2It&#039;s possible to play games by solving each equation for the tangent function and then equating the two. What&#039;s nice about that: you get all the x&#039;s on one side and all the y&#039;s on the other and you can immediately reduce the order of the equation by one (corresponding to the constant solution above). What&#039;s not nice about that: you&#039;re down to only one equation, an equation you cannot directly integrate even if you try to do something fancy like make one side look like the derivative of a quotient. While the coupling of these ODE&#039;s looks a bit weak, it&#039;s still strong enough to resist analytical techniques.

Help with coupled differential equations. I have these two coupled differential equations a ddot{x}+ b dddot{x}+c tan (x+y)dot{x}=0,

Answered question

Answer & Explanation

New Questions in Calculus 2