Internal problem ID [9483]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1153.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{2}-v \left (v -1\right )\right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (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)
\[ y \left (x \right ) = \left (\operatorname {BesselJ}\left (v -\frac {1}{2}, x \sqrt {a}\right ) c_{1} +\operatorname {BesselY}\left (v -\frac {1}{2}, x \sqrt {a}\right ) c_{2} \right ) \sqrt {x} \]
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 44
DSolve[((1 - v)*v + a*x^2)*y[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \sqrt {x} \left (c_1 \operatorname {BesselJ}\left (v-\frac {1}{2},\sqrt {a} x\right )+c_2 \operatorname {BesselY}\left (v-\frac {1}{2},\sqrt {a} x\right )\right ) \]