54.3.156 problem 1170

Internal problem ID [12451]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1170
Date solved : Wednesday, October 01, 2025 at 01:45:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 43

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

\[ y = \frac {c_1 \operatorname {BesselJ}\left (\frac {\sqrt {-4 b +1}}{2}, \sqrt {a}\, x \right )+c_2 \operatorname {BesselY}\left (\frac {\sqrt {-4 b +1}}{2}, \sqrt {a}\, x \right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 58

ode=(b + a*x^2)*y[x] + 2*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 c_1 j_{\frac {1}{2} \left (\sqrt {1-4 b}-1\right )}\left (\sqrt {a} x\right )+c_2 y_{\frac {1}{2} \left (\sqrt {1-4 b}-1\right )}\left (\sqrt {a} x\right ) \end{align*}

✓ Sympy. Time used: 0.165 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = \frac {C_{1} J_{\sqrt {\frac {1}{4} - b}}\left (\sqrt {a} x\right ) + C_{2} Y_{\sqrt {\frac {1}{4} - b}}\left (\sqrt {a} x\right )}{\sqrt {x}} \]