60.3.149 problem 1153

Internal problem ID [11159]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1153
Date solved : Monday, January 27, 2025 at 10:47:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 31

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

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

✓ Solution by Mathematica

Time used: 0.036 (sec). Leaf size: 44

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

\[ y(x)\to \sqrt {x} \left (c_1 \operatorname {BesselJ}\left (v-\frac {1}{2},\sqrt {a} x\right )+c_2 \operatorname {BesselY}\left (v-\frac {1}{2},\sqrt {a} x\right )\right ) \]