60.3.214 problem 1218

Internal problem ID [11224]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1218
Date solved : Tuesday, January 28, 2025 at 05:42:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 26

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

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

✓ Solution by Mathematica

Time used: 0.101 (sec). Leaf size: 37

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

\[ y(x)\to \csc (x) \left (c_1 \operatorname {BesselJ}\left (i \sqrt {a},x\right )+c_2 \operatorname {BesselY}\left (i \sqrt {a},x\right )\right ) \]