I need to solve the equation below with Euler's method: y ″ + π y e x / 3 ( 2 y ′ sin ⁡ ( π x ) + π y cos ⁡ ( π x ) ) = y 9 for the initial conditions y ( 0 ) = 1, y ′ ( 0 ) = − 1 / 3So I know I need to turn the problem into a system of two first order differential equations.Therefore u 1 = y ′ and u 2 = y ″ I can now write the system as: u 1 = y ′ u 2 = y 9 − π y e x / 3 ( 2 u 1 sin ⁡ ( π x ) − π y cos ⁡ ( π x ) )How do I proceed from here?

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I need to solve the equation below with Euler&#039;s method:      y    ″    +  π  y      e          x              /            3        (  2      y    ′    sin  ⁡  (  π  x  )  +  π  y  cos  ⁡  (  π  x  )  )  =      y    9  for the initial conditions   y  (  0  )  =  1,       y    ′    (  0  )  =  −  1      /    3So I know I need to turn the problem into a system of two first order differential equations.Therefore       u    1    =      y    ′   and       u    2    =      y    ″   I can now write the system as:      u    1    =      y    ′          u    2    =            y      9        −  π  y      e          x              /            3        (  2      u    1    sin  ⁡  (  π  x  )  −  π  y  cos  ⁡  (  π  x  )  )How do I proceed from here?

Tristan Pittman · Accepted Answer

Letting   u  =      y    ′   is the right idea. This gives you the pair of equations                              u          ′                                    =                  y          9                −        π        y                  e                      x                          /                        3                          (        2        u        sin        ⁡        (        π        x        )        +        π        y        cos        ⁡        (        π        x        )        )                                      y          ′                                    =        u            which is a standard initial value problem. Notice that there are no   x derivatives, so you can integrate each value of   x separately. So fix some value for   x and discretize the system using your favorite method -- I am a big fan of Velocity Verlet:                              y                      n            +            1                                              =                  y          n                +        h                  u          n                                              u                      n            +            1                                              =                  u          n                +        h                  (                                    y                              n                +                1                                      9                    −          π                      y                          n              +              1                                            e                          x                              /                            3                                (          2                      u            n                    sin          ⁡          (          π          x          )          +          π                      y                          n              +              1                                cos          ⁡          (          π          x          )          )          )                ,            where   h is the time step; you could also use e.g. forward or backward Euler (you say you want &quot;Euler&#039;s method&quot; but there are at least three!). You know       u    0   and       v    0   from the initial conditions, so just iteratively apply the above rules to trace out the trajectory through time.EDIT: Forward Euler would be                              u                      n            +            1                                              =                  u          n                +        h                  (                                    y                              n                                      9                    −          π                      y                          n                                            e                          x                              /                            3                                (          2                      u            n                    sin          ⁡          (          π          x          )          +          π                      y                          n                                cos          ⁡          (          π          x          )          )          )                ,                                      y                      n            +            1                                              =                  y          n                +        h                  u                      n            +            1                              Notice that I strongly recommend against using Explicit Euler in pretty much any circumstance.

y′′+pi ye^(x/3)(2y′sin(pi x)+pi ycos(pi x))=y/9 for the initial conditions y(0)=1, y′(0)=−1/3 solve the equation below with Euler's method

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