54.7.67 problem 1675 (book 6.84)

Internal problem ID [12916]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1675 (book 6.84)
Date solved : Wednesday, October 01, 2025 at 02:46:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 25

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

\[ y = x^{a} \left (c_1 \sin \left (x^{b} c \right )+c_2 \cos \left (x^{b} c \right )\right ) \]

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 69

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

\begin{align*} y(x)&\to 2^{-\frac {a+b}{b}} c^{a/b} \left (x^{2 b}\right )^{\frac {a}{2 b}} \left (2 c_1 \cos \left (c \sqrt {x^{2 b}}\right )+c_2 \sin \left (c \sqrt {x^{2 b}}\right )\right ) \end{align*}

✗ Sympy

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

TypeError : invalid input: 2*a + b