54.3.391 problem 1408

Internal problem ID [12686]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1408
Date solved : Wednesday, October 01, 2025 at 02:19:56 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (x^{2} \left (\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right )+\left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )+\left (x^{2}-\operatorname {a3} \right ) \left (x^{2}-\operatorname {a1} \right )\right )-\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )\right ) y^{\prime }}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )}-\frac {\left (A \,x^{2}+B \right ) y}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )} \end{align*}

✗ Maple

ode:=diff(diff(y(x),x),x) = -(x^2*((x^2-a1)*(x^2-a2)+(x^2-a2)*(x^2-a3)+(x^2-a3)*(x^2-a1))-(x^2-a1)*(x^2-a2)*(x^2-a3))/x/(x^2-a1)/(x^2-a2)/(x^2-a3)*diff(y(x),x)-(A*x^2+B)/x/(x^2-a1)/(x^2-a2)/(x^2-a3)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[x],{x,2}] == -(((B + A*x^2)*y[x])/(x*(-a1 + x^2)*(-a2 + x^2)*(-a3 + x^2))) - (((a1 - x^2)*(-a2 + x^2)*(-a3 + x^2) + x^2*((-a1 + x^2)*(-a2 + x^2) + (-a1 + x^2)*(-a3 + x^2) + (-a2 + x^2)*(-a3 + x^2)))*D[y[x],x])/(x*(-a1 + x^2)*(-a2 + x^2)*(-a3 + x^2)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
B = symbols("B") 
a1 = symbols("a1") 
a2 = symbols("a2") 
a3 = symbols("a3") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + (A*x**2 + B)*y(x)/(x*(-a1 + x**2)*(-a2 + x**2)*(-a3 + x**2)) + (x**2*((-a1 + x**2)*(-a2 + x**2) + (-a1 + x**2)*(-a3 + x**2) + (-a2 + x**2)*(-a3 + x**2)) - (-a1 + x**2)*(-a2 + x**2)*(-a3 + x**2))*Derivative(y(x), x)/(x*(-a1 + x**2)*(-a2 + x**2)*(-a3 + x**2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False