60.3.303 problem 1309

Internal problem ID [11313]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1309
Date solved : Tuesday, January 28, 2025 at 05:58:13 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime }-\left (x^{2}-1\right ) y^{\prime }+y x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.148 (sec). Leaf size: 65

dsolve(x^3*diff(diff(y(x),x),x)-(x^2-1)*diff(y(x),x)+x*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{\frac {1}{4 x^{2}}} \left (c_{1} \left (2 x^{2}-1\right ) \operatorname {BesselI}\left (0, \frac {1}{4 x^{2}}\right )+\left (2 x^{2}-1\right ) c_{2} \operatorname {BesselK}\left (0, -\frac {1}{4 x^{2}}\right )+\operatorname {BesselI}\left (1, \frac {1}{4 x^{2}}\right ) c_{1} +\operatorname {BesselK}\left (1, -\frac {1}{4 x^{2}}\right ) c_{2} \right )}{x} \]

✓ Solution by Mathematica

Time used: 0.225 (sec). Leaf size: 77

DSolve[x*y[x] - (-1 + x^2)*D[y[x],x] + x^3*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 G_{1,2}^{2,0}\left (-\frac {1}{2 x^2}| \begin {array}{c} 1 \\ -\frac {1}{2},-\frac {1}{2} \\ \end {array} \right )+\frac {c_1 e^{\frac {1}{4 x^2}} \left (\left (2 x^2-1\right ) \operatorname {BesselI}\left (0,\frac {1}{4 x^2}\right )+\operatorname {BesselI}\left (1,\frac {1}{4 x^2}\right )\right )}{\sqrt {2} x} \]