54.3.137 problem 1151

Internal problem ID [12432]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1151
Date solved : Wednesday, October 01, 2025 at 01:44:57 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 42

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

\[ y = \frac {c_2 \,{\mathrm e}^{-\sqrt {a}\, x} \left (a x +\sqrt {a}\right )+c_1 \,{\mathrm e}^{\sqrt {a}\, x} \left (a x -\sqrt {a}\right )}{x} \]

✓ Mathematica. Time used: 0.05 (sec). Leaf size: 88

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

\begin{align*} y(x)&\to -\frac {i \sqrt {\frac {2}{\pi }} \sqrt {x} \left (\left (i \sqrt {a} c_2 x+c_1\right ) \sinh \left (\sqrt {a} x\right )-\left (\sqrt {a} c_1 x+i c_2\right ) \cosh \left (\sqrt {a} x\right )\right )}{\left (-i \sqrt {a} x\right )^{3/2}} \end{align*}

✓ Sympy. Time used: 0.051 (sec). Leaf size: 34

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

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