60.3.185 problem 1189

Internal problem ID [11195]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1189
Date solved : Monday, January 27, 2025 at 10:48:27 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.096 (sec). Leaf size: 80

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

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

✓ Solution by Mathematica

Time used: 0.161 (sec). Leaf size: 168

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

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