60.3.16 problem 1016

Internal problem ID [11026]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1016
Date solved : Tuesday, January 28, 2025 at 05:40:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.961 (sec). Leaf size: 85

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

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

✓ Solution by Mathematica

Time used: 0.239 (sec). Leaf size: 225

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

\[ y(x)\to 2^{\frac {c}{2 c+2}} x^{-c/2} \left (x^{c+1}\right )^{\frac {c}{2 c+2}} e^{-\frac {\sqrt {a} x^{c+1}}{\sqrt {-(c+1)^2}}} \left (c_1 \operatorname {HypergeometricU}\left (-\frac {(c+1) \left (c b+b+\sqrt {a} c \sqrt {-(c+1)^2}\right )}{2 \sqrt {a} \left (-(c+1)^2\right )^{3/2}},\frac {c}{c+1},\frac {2 \sqrt {a} x^{c+1}}{\sqrt {-(c+1)^2}}\right )+c_2 L_{\frac {(c+1) \left (c b+b+\sqrt {a} c \sqrt {-(c+1)^2}\right )}{2 \sqrt {a} \left (-(c+1)^2\right )^{3/2}}}^{-\frac {1}{c+1}}\left (\frac {2 \sqrt {a} x^{c+1}}{\sqrt {-(c+1)^2}}\right )\right ) \]