60.3.163 problem 1167

Internal problem ID [11173]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1167
Date solved : Monday, January 27, 2025 at 10:47:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.090 (sec). Leaf size: 63

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

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

✓ Solution by Mathematica

Time used: 0.131 (sec). Leaf size: 130

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

\[ y(x)\to m^{-2/m} a^{\frac {1}{m}} \left (x^m\right )^{\frac {1}{m}} \left (c_1 \operatorname {Gamma}\left (1-\frac {2 i \sqrt {b-1}}{m}\right ) \operatorname {BesselJ}\left (-\frac {2 i \sqrt {b-1}}{m},\frac {2 \sqrt {a} \sqrt {x^m}}{m}\right )+c_2 \operatorname {Gamma}\left (\frac {2 i \sqrt {b-1}}{m}+1\right ) \operatorname {BesselJ}\left (\frac {2 i \sqrt {b-1}}{m},\frac {2 \sqrt {a} \sqrt {x^m}}{m}\right )\right ) \]