54.3.195 problem 1209

Internal problem ID [12490]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1209
Date solved : Wednesday, October 01, 2025 at 01:45:47 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 38

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

\[ y = \frac {\left (\pi \,\operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right ) c_2 +c_1 \right ) {\mathrm e}^{-\frac {x^{2}}{2}}-i \sqrt {\pi }\, \sqrt {2}\, c_2 x}{x^{2}} \]

✓ Mathematica. Time used: 0.076 (sec). Leaf size: 61

ode=(-2 + x^2)*y[x] + x*(2 + x^2)*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 \frac {-\sqrt {2 \pi } c_1 e^{-\frac {x^2}{2}} \text {erfi}\left (\frac {x}{\sqrt {2}}\right )+2 c_2 e^{-\frac {x^2}{2}-2}+2 c_1 x}{2 x^2} \end{align*}

✗ Sympy

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

False