60.3.168 problem 1172

Internal problem ID [11178]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1172
Date solved : Tuesday, January 28, 2025 at 05:41:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+2 \left (x -1\right ) y^{\prime }+a y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.060 (sec). Leaf size: 47

dsolve(x^2*diff(diff(y(x),x),x)+2*(x-1)*diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
 

\[ y = \sqrt {\frac {1}{x}}\, {\mathrm e}^{-\frac {1}{x}} \left (c_{1} \operatorname {BesselI}\left (\frac {\sqrt {-4 a +1}}{2}, \frac {1}{x}\right )+c_{2} \operatorname {BesselK}\left (\frac {\sqrt {-4 a +1}}{2}, \frac {1}{x}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.146 (sec). Leaf size: 145

DSolve[a*y[x] + 2*(-1 + x)*D[y[x],x] + x^2*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to 2^{\frac {1}{2}-\frac {1}{2} \sqrt {1-4 a}} \left (\frac {1}{x}\right )^{\frac {1}{2}-\frac {1}{2} \sqrt {1-4 a}} \left (2^{\sqrt {1-4 a}} c_2 \left (\frac {1}{x}\right )^{\sqrt {1-4 a}} \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (\sqrt {1-4 a}+1\right ),\sqrt {1-4 a}+1,-\frac {2}{x}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {1}{2}-\frac {1}{2} \sqrt {1-4 a},1-\sqrt {1-4 a},-\frac {2}{x}\right )\right ) \]