54.3.88 problem 1102

Internal problem ID [12383]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1102
Date solved : Wednesday, October 01, 2025 at 01:44:11 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }+2 y^{\prime }+a \,x^{2} y&=0 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 33

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

\[ y = \frac {c_1 \operatorname {BesselJ}\left (\frac {1}{3}, \frac {2 \sqrt {a}\, x^{{3}/{2}}}{3}\right )+c_2 \operatorname {BesselY}\left (\frac {1}{3}, \frac {2 \sqrt {a}\, x^{{3}/{2}}}{3}\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 36

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

\begin{align*} y(x)&\to \frac {c_1 \operatorname {AiryAi}\left (\sqrt [3]{-a} x\right )+c_2 \operatorname {AiryBi}\left (\sqrt [3]{-a} x\right )}{x} \end{align*}

✓ Sympy. Time used: 0.120 (sec). Leaf size: 44

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

\[ y{\left (x \right )} = \frac {C_{1} J_{\frac {1}{3}}\left (\frac {2 \sqrt {a} x^{\frac {3}{2}}}{3}\right ) + C_{2} Y_{\frac {1}{3}}\left (\frac {2 \sqrt {a} x^{\frac {3}{2}}}{3}\right )}{\sqrt {x}} \]