Internal problem ID [8786]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1206.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x^{2}+\left (2 a x +b \right ) x y^{\prime }+\left (a b x +c \,x^{2}+d \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 87
dsolve(x^2*diff(diff(y(x),x),x)+(2*a*x+b)*x*diff(y(x),x)+(a*b*x+c*x^2+d)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-a x} x^{-\frac {b}{2}+\frac {1}{2}} \BesselJ \left (\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}, \sqrt {-a^{2}+c}\, x \right )+c_{2} {\mathrm e}^{-a x} x^{-\frac {b}{2}+\frac {1}{2}} \BesselY \left (\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}, \sqrt {-a^{2}+c}\, x \right ) \]
✓ Solution by Mathematica
Time used: 0.041 (sec). Leaf size: 98
DSolve[(d + a*b*x + c*x^2)*y[x] + x*(b + 2*a*x)*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-a x} x^{\frac {1}{2}-\frac {b}{2}} \left (c_1 J_{\frac {1}{2} \sqrt {(b-1)^2-4 d}}\left (-i \sqrt {a^2-c} x\right )+c_2 Y_{\frac {1}{2} \sqrt {(b-1)^2-4 d}}\left (-i \sqrt {a^2-c} x\right )\right ) \\ \end{align*}