60.3.165 problem 1179

Internal problem ID [11161]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1179
Date solved : Sunday, March 30, 2025 at 07:44:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 19

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

\[ y = x \left (c_1 \sin \left (a x \right )+c_2 \cos \left (a x \right )\right ) \]

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 38

ode=(2 + a^2*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]
 

\[ y(x)\to c_1 x e^{-i a x}-\frac {i c_2 x e^{i a x}}{2 a} \]

✓ Sympy. Time used: 0.240 (sec). Leaf size: 24

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

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