60.3.202 problem 1206

Internal problem ID [11212]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1206
Date solved : Tuesday, January 28, 2025 at 05:41:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (2 a x +b \right ) x y^{\prime }+\left (a b x +c \,x^{2}+d \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 76

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)
 

\[ y = x^{-\frac {b}{2}+\frac {1}{2}} {\mathrm e}^{-a x} \left (c_{1} \operatorname {BesselJ}\left (\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}, \sqrt {-a^{2}+c}\, x \right )+c_{2} \operatorname {BesselY}\left (\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}, \sqrt {-a^{2}+c}\, x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.149 (sec). Leaf size: 110

DSolve[(d + a*b*x + c*x^2)*y[x] + x*(b + 2*a*x)*D[y[x],x] + x^2*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (c_1 \operatorname {BesselJ}\left (\frac {1}{2} \sqrt {b^2-2 b-4 d+1},-i \sqrt {a^2-c} x\right )+c_2 \operatorname {BesselY}\left (\frac {1}{2} \sqrt {b^2-2 b-4 d+1},-i \sqrt {a^2-c} x\right )\right ) \exp \left (\int _1^x-\frac {b+2 a K[1]-1}{2 K[1]}dK[1]\right ) \]