Euler differential equationsI have following three equations u ″ − 2 u = − 2 v u ( 0 ) = 0 u ′ ( 1 ) = 0and from these 3 equations I am trying to find u(v).It looks to me "Cauchy-Euler Differential Equations - Nonhomogeneous case" but I am not sure about that because it is not an exactly Cauchy form. Could you help me to figure out u(v)?

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Euler differential equationsI have following three equations      u    ″    −  2  u  =  −  2  v  u  (  0  )  =  0      u    ′    (  1  )  =  0and from these 3 equations I am trying to find u(v).It looks to me &quot;Cauchy-Euler Differential Equations - Nonhomogeneous case&quot; but I am not sure about that because it is not an exactly Cauchy form. Could you help me to figure out u(v)?

enreciarpv · Accepted Answer

Step 1The solution to the homogeneous case is      u    c    (  v  )  =      c          1            e                  2            v        +      c          2            e          −              2            v       and by inspection, we observe that a particular solution is       u    p    (  v  )  =  v so the solution to the general case is  u  (  v  )  =      c          1            e                  2            v        +      c          2            e          −              2            v        +  v. Then,  u  (  0  )  =  0  ⇒      c    1    +      c    2    =  0      u    ′    (  1  )  =  0  ⇒      2        c          1            e                  2              −      2        c          2            e          −              2              +  1  =  0Step 2from which      c    1    =                    −        1                              2                          e                                    2                                      +                  2                          e                      −                          2                                              and      c    2    =            1                        2                          e                                    2                                      +                  2                          e                      −                          2                                          Now   u  (  v  )  =      u    c    (  v  )  +      u    p    (  v  )

Euler differential equations. I have following three equations u′′-2u=-2v

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