60.3.193 problem 1197

Internal problem ID [11203]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1197
Date solved : Tuesday, January 28, 2025 at 05:41:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.082 (sec). Leaf size: 43

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

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

✓ Solution by Mathematica

Time used: 0.090 (sec). Leaf size: 76

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

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