60.3.151 problem 1155

Internal problem ID [11161]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1155
Date solved : Monday, January 27, 2025 at 10:47:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.101 (sec). Leaf size: 71

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

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

✓ Solution by Mathematica

Time used: 0.101 (sec). Leaf size: 116

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

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