Internal problem ID [8787]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1207.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime } x +\left (\mathit {a1} \,x^{2}+\mathit {b1} x +\mathit {c1} \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.052 (sec). Leaf size: 119
dsolve(x^2*diff(diff(y(x),x),x)+(a*x+b)*diff(y(x),x)*x+(a1*x^2+b1*x+c1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-\frac {a x}{2}} x^{-\frac {b}{2}} \WhittakerM \left (-\frac {a b -2 \mathit {b1}}{2 \sqrt {a^{2}-4 \mathit {a1}}}, \frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}, \sqrt {a^{2}-4 \mathit {a1}}\, x \right )+c_{2} {\mathrm e}^{-\frac {a x}{2}} x^{-\frac {b}{2}} \WhittakerW \left (-\frac {a b -2 \mathit {b1}}{2 \sqrt {a^{2}-4 \mathit {a1}}}, \frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}, \sqrt {a^{2}-4 \mathit {a1}}\, x \right ) \]
✓ Solution by Mathematica
Time used: 0.052 (sec). Leaf size: 213
DSolve[(c1 + b1*x + a1*x^2)*y[x] + x*(b + a*x)*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-\frac {1}{2} x \left (\sqrt {a^2-4 \text {a1}}+a\right )} x^{\frac {1}{2} \left (\sqrt {(b-1)^2-4 \text {c1}}-b+1\right )} \left (c_1 \text {HypergeometricU}\left (\frac {\sqrt {a^2-4 \text {a1}} \left (\sqrt {(b-1)^2-4 \text {c1}}+1\right )+a b-2 \text {b1}}{2 \sqrt {a^2-4 \text {a1}}},\sqrt {(b-1)^2-4 \text {c1}}+1,x \sqrt {a^2-4 \text {a1}}\right )+c_2 L_{\frac {-a b+2 \text {b1}-\sqrt {a^2-4 \text {a1}} \left (\sqrt {(b-1)^2-4 \text {c1}}+1\right )}{2 \sqrt {a^2-4 \text {a1}}}}^{\sqrt {(b-1)^2-4 \text {c1}}}\left (\sqrt {a^2-4 \text {a1}} x\right )\right ) \\ \end{align*}