Internal problem ID [8882]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1303.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (d x +f \right ) y^{\prime }+g y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.028 (sec). Leaf size: 501
dsolve((a*x^2+b*x+c)*diff(diff(y(x),x),x)+(d*x+f)*diff(y(x),x)+g*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [\frac {-a +d +\sqrt {a^{2}+\left (-2 d -4 g \right ) a +d^{2}}}{2 a}, -\frac {a -d +\sqrt {a^{2}+\left (-2 d -4 g \right ) a +d^{2}}}{2 a}\right ], \left [\frac {d \sqrt {\frac {-4 c a +b^{2}}{a^{2}}}\, a -2 a f +b d}{2 a^{2} \sqrt {\frac {-4 c a +b^{2}}{a^{2}}}}\right ], \frac {\left (-2 a^{2} x -a b \right ) \sqrt {\frac {-4 c a +b^{2}}{a^{2}}}+4 c a -b^{2}}{8 c a -2 b^{2}}\right )+c_{2} \left (2 \sqrt {\frac {-4 c a +b^{2}}{a^{2}}}\, x \,a^{2}+\sqrt {\frac {-4 c a +b^{2}}{a^{2}}}\, b a -4 c a +b^{2}\right )^{\frac {a \left (a -\frac {d}{2}\right ) \sqrt {\frac {-4 c a +b^{2}}{a^{2}}}+a f -\frac {b d}{2}}{\sqrt {\frac {-4 c a +b^{2}}{a^{2}}}\, a^{2}}} \hypergeom \left (\left [\frac {a \left (a -\sqrt {a^{2}+\left (-2 d -4 g \right ) a +d^{2}}\right ) \sqrt {\frac {-4 c a +b^{2}}{a^{2}}}+2 a f -b d}{2 \sqrt {\frac {-4 c a +b^{2}}{a^{2}}}\, a^{2}}, \frac {a \left (a +\sqrt {a^{2}+\left (-2 d -4 g \right ) a +d^{2}}\right ) \sqrt {\frac {-4 c a +b^{2}}{a^{2}}}+2 a f -b d}{2 \sqrt {\frac {-4 c a +b^{2}}{a^{2}}}\, a^{2}}\right ], \left [\frac {4 a^{2} \sqrt {\frac {-4 c a +b^{2}}{a^{2}}}-d \sqrt {\frac {-4 c a +b^{2}}{a^{2}}}\, a +2 a f -b d}{2 a^{2} \sqrt {\frac {-4 c a +b^{2}}{a^{2}}}}\right ], \frac {\left (-2 a^{2} x -a b \right ) \sqrt {\frac {-4 c a +b^{2}}{a^{2}}}+4 c a -b^{2}}{8 c a -2 b^{2}}\right ) \]
✓ Solution by Mathematica
Time used: 2.965 (sec). Leaf size: 498
DSolve[g*y[x] + (f + d*x)*y'[x] + (c + b*x + a*x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \, _2F_1\left (-\frac {a-d+\sqrt {(a-d)^2-4 a g}}{2 a},\frac {-a+d+\sqrt {(a-d)^2-4 a g}}{2 a};\frac {\left (b+\sqrt {b^2-4 a c}\right ) d-2 a f}{2 a \sqrt {b^2-4 a c}};\frac {b+2 a x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )-c_2 2^{\frac {\frac {b d}{\sqrt {b^2-4 a c}}+d}{2 a}-\frac {f}{\sqrt {b^2-4 a c}}-1} \exp \left (-\frac {i \pi \left (d \left (\sqrt {b^2-4 a c}+b\right )-2 a f\right )}{2 a \sqrt {b^2-4 a c}}\right ) \left (\frac {\sqrt {b^2-4 a c}+2 a x+b}{\sqrt {b^2-4 a c}}\right )^{-\frac {\frac {b d}{\sqrt {b^2-4 a c}}+d}{2 a}+\frac {f}{\sqrt {b^2-4 a c}}+1} \, _2F_1\left (\frac {\frac {2 f a}{\sqrt {b^2-4 a c}}+a-\frac {b d}{\sqrt {b^2-4 a c}}-\sqrt {(a-d)^2-4 a g}}{2 a},\frac {\frac {2 f a}{\sqrt {b^2-4 a c}}+a-\frac {b d}{\sqrt {b^2-4 a c}}+\sqrt {(a-d)^2-4 a g}}{2 a};-\frac {\frac {b d}{\sqrt {b^2-4 a c}}+d+a \left (-\frac {2 f}{\sqrt {b^2-4 a c}}-4\right )}{2 a};\frac {b+2 a x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right ) \\ \end{align*}