54.3.388 problem 1405

Internal problem ID [12683]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1405
Date solved : Wednesday, October 01, 2025 at 02:19:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {\left (a \,x^{4}+10 x^{2}+1\right ) y}{4 x^{6}} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 41

ode:=diff(diff(y(x),x),x) = (2*x^2+1)/x^3*diff(y(x),x)-1/4*(a*x^4+10*x^2+1)/x^6*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {1}{4 x^{2}}} x^{{3}/{2}} \left (x^{\frac {\sqrt {-a +9}}{2}} c_1 +x^{-\frac {\sqrt {-a +9}}{2}} c_2 \right ) \]

✓ Mathematica. Time used: 0.131 (sec). Leaf size: 74

ode=D[y[x],{x,2}] == -1/4*((1 + 10*x^2 + a*x^4)*y[x])/x^6 + ((1 + 2*x^2)*D[y[x],x])/x^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^{-\frac {1}{4 x^2}-\frac {3}{2}} x^{\frac {3}{2}-\frac {\sqrt {9-a}}{2}} \left (c_2 x^{\sqrt {9-a}}+\sqrt {9-a} c_1\right )}{\sqrt {9-a}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) - (2*x**2 + 1)*Derivative(y(x), x)/x**3 + (a*x**4 + 10*x**2 + 1)*y(x)/(4*x**6),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (a*x**4*y(x) + 4*x**6*Derivative(y(x), (x, 2)) + 10*x**2*y(x) + y(x))/(8*x**5 + 4*x**3) cannot be solved by the factorable group method