60.3.196 problem 1200

Internal problem ID [11206]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1200
Date solved : Tuesday, January 28, 2025 at 05:41:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 27

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

\[ y = \sqrt {x}\, {\mathrm e}^{-x} \left (c_{1} \operatorname {BesselI}\left (-\frac {1}{2}+v , x\right )+c_{2} \operatorname {BesselK}\left (-\frac {1}{2}+v , x\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.046 (sec). Leaf size: 45

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

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