60.3.270 problem 1275

Internal problem ID [11280]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1275
Date solved : Tuesday, January 28, 2025 at 05:57:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (-x^{2}+2 \left (1-m +2 l \right ) x -m^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.276 (sec). Leaf size: 53

dsolve(4*x^2*diff(diff(y(x),x),x)+4*x*diff(y(x),x)+(-x^2+2*(1-m+2*l)*x-m^2+1)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.079 (sec). Leaf size: 97

DSolve[(1 - m^2 + 2*(1 + 2*l - m)*x - x^2)*y[x] + 4*x*D[y[x],x] + 4*x^2*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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