I am studying about the linear odes with non-constant coefficients.I know the first order linear ode with non-constant coefficient (1) y ′ ( x ) + f ( x ) y ( x ) = 0 has a general solution of the form (2) y = C e − ∫ f ( x ) d x However, I am more interested in the case of linear second order odes with non-constant coefficients (3) y ″ ( x ) + g ( x ) y ′ ( x ) + f ( x ) y ( x ) = 0 I know that this equation does not have a closed form solution like (2). However, I am interested in special cases of that.Questions1. Consider (3), when g ( x ) = 0, then we have (4) y ″ ( x ) + f ( x ) y ( x ) = 0 Is Eq.(4) a famous well-known equation? If YES, what is its name?2. Does (4) have a closed form solution like (2)?3. Can you name or give me a list of well-known linear second order odes with non-constant coefficients which are not polynomial?For example, I know Cauchy-Euler, Airy, Bessel, Chebyshev, Laguerre and Legendre equations whose coefficients are polynomials. But I don't know any well-known equation with non-polynomial coefficients.

Question

I am studying about the linear odes with non-constant coefficients.I know the first order linear ode with non-constant coefficient                    (1)                              y                                                  ′                                      (        x        )        +        f        (        x        )        y        (        x        )        =        0            has a general solution of the form                    (2)                    y        =        C                  e                      −            ∫            f            (            x            )            d            x                              However, I am more interested in the case of linear second order odes with non-constant coefficients                    (3)                              y                                                  ″                                      (        x        )        +        g        (        x        )                  y                                                  ′                                      (        x        )        +        f        (        x        )        y        (        x        )        =        0            I know that this equation does not have a closed form solution like (2). However, I am interested in special cases of that.Questions1. Consider (3), when   g  (  x  )  =  0, then we have                    (4)                              y                                                  ″                                      (        x        )        +        f        (        x        )        y        (        x        )        =        0            Is Eq.(4) a famous well-known equation? If YES, what is its name?2. Does (4) have a closed form solution like (2)?3. Can you name or give me a list of well-known linear second order odes with non-constant coefficients which are not polynomial?For example, I know Cauchy-Euler, Airy, Bessel, Chebyshev, Laguerre and Legendre equations whose coefficients are polynomials. But I don&#039;t know any well-known equation with non-polynomial coefficients.

Ronnie Glenn · Accepted Answer

Somewhere (but where ?) I already answer to a question quite the same as your question 1 (only change a sign in equation 4). By luck, I didn&#039;t remove my draft (copy below).There is no general formula for the solutions of equation 4 in cases of any f(x) since specific special functions are defined according to each specific case. All the more so for equation 3.COPY :      y    ″    (  x  )  =  f  (  x  )  y  (  x  )Case:  f  (  x  )  =      c    2      →    y  (  x  )  =      c    1        e          c      x        +      c    2        e          −      c      x        =      c    3    cosh  ⁡  (  c  x  )  +      c    4    sinh  ⁡  (  c  x  )Case:   f  (  x  )  =  −      c    2      →    y  (  x  )  =      c    1    cos  ⁡  (  c  x  )  +      c    2    sin  ⁡  (  c  x  )Case:   f  (  x  )  =  x    →    y  (  x  )  =      c    1    A  i  (  x  )  +      c    2    B  i  (  x  )  . Ary functions.Case:   f  (  x  )  =      x    2      →    y  (  x  )  =      c    1        D          −      1              /            2        (      2    x  )  +      c    2        D          −      1              /            2        (  i      2    x  )  . Parabolic cylinder function.Case:   f  (  x  )  =  −      λ    2        x                  1        ν            −      2          →    y  (  x  )  =      c    1        x        J          ν        (  λ  x  )  +      c    2        x        Y          ν        (  λ  x  )  . Bessel functionsCase:  f  (  x  )  =      λ    2        x                  1        ν            −      2          →    y  (  x  )  =      c    1        x        I          ν        (  λ  x  )  +      c    2        x        K          ν        (  λ  x  )  . Modified Bessel functions.Case:   f  (  x  )  =  −  a  +  2  b  cos  ⁡  (  2  x  )    →    y  (  x  )  =      c    1    C  (  a    ,    b    ;    x  )  +      c    1    S  (  a    ,    b    ;    x  ). Mathieu functions.Case:   f  (  x  )  =      A          x      2        +      B    x    +  C    →    y  (  x  )  =      e          −              γ        2            x            x                  β        2                  (          c      1        M    (    α        ,        β        ;        γ    x    )    +          c      2        U    (    α        ,        β        ;        γ    x    )    )     with  f  (  x  )  =      A          x      2        +      B    x    +  C    →    y  (  x  )  =      e          −              γ        2            x            x                  β        2                  (          c      1        M    (    α        ,        β        ;        γ    x    )    +          c      2        U    (    α        ,        β        ;        γ    x    )    )          {                            γ          =          ±          2                      C                                                β          =          1          ±          2                      A            +                          1              4                                                            α          =                      β            2                    +                      B            γ                                    Kummer and Tricomi functions (confluent hypergeometric functions).Etc.

I am studying about the linear odes with non-constant coefficients. I know the first order linear o

Answered question

Answer & Explanation

New Questions in Linear algebra