60.3.378 problem 1384

Internal problem ID [11388]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1384
Date solved : Tuesday, January 28, 2025 at 06:04:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (-x^{2} \left (a^{2}-1\right )+2 \left (a +3\right ) b x -b^{2}\right ) y}{4 x^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.145 (sec). Leaf size: 73

dsolve(diff(diff(y(x),x),x) = -1/4*(-x^2*(a^2-1)+2*(a+3)*b*x-b^2)/x^2*y(x),y(x), singsol=all)
 

\[ y = c_{1} \operatorname {WhittakerM}\left (\frac {b \left (a +3\right )}{2 \sqrt {a^{2}-1}}, \frac {\sqrt {b^{2}+1}}{2}, \sqrt {a^{2}-1}\, x \right )+c_{2} \operatorname {WhittakerW}\left (\frac {b \left (a +3\right )}{2 \sqrt {a^{2}-1}}, \frac {\sqrt {b^{2}+1}}{2}, \sqrt {a^{2}-1}\, x \right ) \]

✓ Solution by Mathematica

Time used: 0.051 (sec). Leaf size: 106

DSolve[D[y[x],{x,2}] == -1/4*((-b^2 + 2*(3 + a)*b*x - (-1 + a^2)*x^2)*y[x])/x^2,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 M_{\frac {(a+3) b}{2 \sqrt {a^2-1}},\frac {\sqrt {b^3+b}}{2 \sqrt {b}}}\left (\sqrt {a^2-1} x\right )+c_2 W_{\frac {(a+3) b}{2 \sqrt {a^2-1}},\frac {\sqrt {b^3+b}}{2 \sqrt {b}}}\left (\sqrt {a^2-1} x\right ) \]