60.3.16 problem 1016

Internal problem ID [11012]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1016
Date solved : Sunday, March 30, 2025 at 07:39:05 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*}

✓ Maple. Time used: 0.181 (sec). Leaf size: 85

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

\[ y = x^{-\frac {c}{2}} \left (c_1 \operatorname {WhittakerM}\left (-\frac {i b}{\sqrt {a}\, \left (2+2 c \right )}, \frac {1}{2+2 c}, \frac {2 i \sqrt {a}\, x^{c} x}{c +1}\right )+c_2 \operatorname {WhittakerW}\left (-\frac {i b}{\sqrt {a}\, \left (2+2 c \right )}, \frac {1}{2+2 c}, \frac {2 i \sqrt {a}\, x^{c} x}{c +1}\right )\right ) \]

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

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq((a*x**(2*c) + b*x**(c - 1))*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : Mul object cannot be interpreted as an integer