# Second order homogeneous ordinary differential equation with variable coefficients.

I am trying to solve the following differential equation, $$$$y^{''}_{xx} + a_1 x y^{'}_{x} +(a_0x+b_0)y=0$$$$ This equation is not the Cauchy‐Euler Equidimensional. I found a solution to the following differntial equation in Handbook of exact solutions for ordinary differential equations by Andrei Polyanin and Valentin Zaitsev, $$$$(ax_2+b2) y^{''}_{xx} + (a_1 x +b_1) y^{'}_{x} + (a_0 x + b_0) y=0$$$$ The equation that I am trying to solve is special case of the prevous equation ($$a_2=b_10, b_2=1$$). The solution to this equation is $$$$y=e^{hx} z(\zeta), where \, \zeta=\dfrac{x-\mu}{\lambda}$$$$ The authors mentioned that $$z(\zeta)$$ matches $$\mathcal{L}(a,b;x)$$ arbitrary solution of the degenerate hyperheometric equation $$x y^{''}_{xx}+ (b-x) y^{'}_{x} -ay=0$$, yet it is not clear how to do that. I have attached four pages of the books that refere to that solution. The problem is number 103. Thanks

• Why are you mixing two notations for derivative? Does $y''_{xx}$ just mean $y''$ (where the independent variable is $x$)? Commented Mar 23, 2020 at 3:28
• Yet, they are the same. I just follow the notation in the Andrei Polyanin and Valentin Zaitsev. Commented Mar 23, 2020 at 3:29

According to Maple, your DE has general solution $$y \left( x \right) =c_1 {{\rm e}^{-{\frac {a_{{0}}x}{a_{{1}}}}}} { { U}\left({\frac {b_{{0}}{a_{{1}}}^{2}+{a_{{0}}}^{2}}{2\,{a_{{1}}}^ {3}}},\,{\frac{1}{2}},\,-{\frac { \left( x{a_{{1}}}^{2}-2\,a_{{0}} \right) ^{2}}{2\,{a_{{1}}}^{3}}}\right)}+ c_2{{\rm e}^{-{\frac {a_{{0}}x}{a_{{1}}}}}} {{ M}\left({\frac {b_{{0}}{a_{{1}}}^{2}+{a_{{0}}}^{2}}{2\,{a_{{1}}}^{3}}},\,{\frac{1}{2} },\,-{\frac { \left( x{a_{{1}}}^{2}-2\,a_{{0}} \right) ^{2}}{2\,{a_{{1 }}}^{3}}}\right)}$$ where $$U$$ and $$M$$ are Kummer functions.
• Thanks, a lot, I am wondering if the general solution is just a repetitive of the same terms. $U$ and $M$ look identical. Commented Mar 23, 2020 at 4:15
• $U$ and $M$ are quite different. $M(p, 1/2, t) = 1 + 2 p t + \ldots$ while $U(p,1/2,t) = \frac{\sqrt{\pi}}{\Gamma(p+1/2)} - \frac{2\sqrt{\pi}}{\Gamma(p)} \sqrt{t} + \ldots$. Commented Mar 23, 2020 at 13:37