For some research Im doing, I've derived an equation of the form below for C ( r ) C ″ + 2 r C ′ = W + f C Or, if you prefer, C C ″ + 2 r C C ′ − W ⋅ C = fThis has the form of a 2nd order inhomogeneous linear equation with variable coefficients but the problem for me here is I don't know a clever way to solve this as C appears throughout equation. Any ideas on how to solve this and indeed, if there is even a solution ?Id plug it into mathematica but am out of office travelling so would be most grateful for input!

Question

For some research Im doing, I&#039;ve derived an equation of the form below for   C  (  r  )      C    ″    +      2    r        C    ′    =  W  +      f    C  Or, if you prefer,  C      C    ″    +      2    r    C      C    ′    −  W  ⋅  C  =  fThis has the form of a 2nd order inhomogeneous linear equation with variable coefficients but the problem for me here is I don&#039;t know a clever way to solve this as C appears throughout equation. Any ideas on how to solve this and indeed, if there is even a solution ?Id plug it into mathematica but am out of office travelling so would be most grateful for input!

Jordan Mcpherson · Accepted Answer

Hint:Let   U  =  r  C,Then                     d        U                    d        r              =  r                    d        C                    d        r              +  C                              d          2                U                    d                  r          2                      =  r                              d          2                C                    d                  r          2                      +                    d        C                    d        r              +                    d        C                    d        r              =  r                              d          2                C                    d                  r          2                      +  2                    d        C                    d        r              ∴            1      r                                    d          2                U                    d                  r          2                      =  W  +                    f        r            U                                    d          2                U                    d                  r          2                      =  W  r  +                    f                  r          2                    U      You can consider as two members Emden-Fowler type nonlinear ODE