Internal problem ID [8881]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1302.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\mathit {A2} \left (a x +b \right )^{2} y^{\prime \prime }+\mathit {A1} \left (a x +b \right ) y^{\prime }+\mathit {A0} \left (a x +b \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 117
dsolve(A2*(a*x+b)^2*diff(diff(y(x),x),x)+A1*(a*x+b)*diff(y(x),x)+A0*(a*x+b)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (a x +b \right )^{-\frac {-a \mathit {A2} +\mathit {A1}}{2 a \mathit {A2}}} \BesselJ \left (\frac {a \mathit {A2} -\mathit {A1}}{a \mathit {A2}}, 2 \sqrt {\mathit {A0}}\, \sqrt {\frac {a x +b}{a^{2} \mathit {A2}}}\right )+c_{2} \left (a x +b \right )^{-\frac {-a \mathit {A2} +\mathit {A1}}{2 a \mathit {A2}}} \BesselY \left (\frac {a \mathit {A2} -\mathit {A1}}{a \mathit {A2}}, 2 \sqrt {\mathit {A0}}\, \sqrt {\frac {a x +b}{a^{2} \mathit {A2}}}\right ) \]
✓ Solution by Mathematica
Time used: 0.041 (sec). Leaf size: 147
DSolve[A0*(b + a*x)*y[x] + A1*(b + a*x)*y'[x] + A2*(b + a*x)^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (\frac {b}{a}+x\right )^{\frac {\text {A1}}{2 a \text {A2}}} (\text {A2} (a x+b))^{-\frac {\text {A1}}{2 a \text {A2}}} \left (c_1 \, _0\tilde {F}_1\left (;\frac {\text {A1}}{a \text {A2}};-\frac {\text {A0} (b+a x)}{a^2 \text {A2}}\right )-c_2 (-1)^{-\frac {\text {A1}}{a \text {A2}}} \left (-\frac {\text {A0} (a x+b)}{a^2 \text {A2}}\right )^{\frac {1}{2}-\frac {\text {A1}}{2 a \text {A2}}} K_{\frac {\text {A1}}{a \text {A2}}-1}\left (2 \sqrt {-\frac {\text {A0} (b+a x)}{a^2 \text {A2}}}\right )\right ) \\ \end{align*}