60.3.167 problem 1171

Internal problem ID [11177]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1171
Date solved : Tuesday, January 28, 2025 at 05:41:43 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.150 (sec). Leaf size: 45

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

\[ y = \frac {c_{1} \operatorname {WhittakerM}\left (-\frac {i a}{2 \sqrt {l}}, n +\frac {1}{2}, 2 i \sqrt {l}\, x \right )+c_{2} \operatorname {WhittakerW}\left (-\frac {i a}{2 \sqrt {l}}, n +\frac {1}{2}, 2 i \sqrt {l}\, x \right )}{x} \]

✓ Solution by Mathematica

Time used: 0.072 (sec). Leaf size: 92

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

\[ y(x)\to e^{-i \sqrt {l} x} x^n \left (c_1 \operatorname {HypergeometricU}\left (\frac {i a}{2 \sqrt {l}}+n+1,2 n+2,2 i \sqrt {l} x\right )+c_2 L_{-\frac {i a}{2 \sqrt {l}}-n-1}^{2 n+1}\left (2 i \sqrt {l} x\right )\right ) \]