59.1.331 problem 338

Internal problem ID [9503]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 338
Date solved : Wednesday, March 05, 2025 at 07:50:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 16 x^{2} y^{\prime \prime }+32 x y^{\prime }+\left (x^{4}-12\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.037 (sec). Leaf size: 25

ode:=16*x^2*diff(diff(y(x),x),x)+32*x*diff(y(x),x)+(x^4-12)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_{1} \sin \left (\frac {x^{2}}{8}\right )+c_{2} \cos \left (\frac {x^{2}}{8}\right )}{x^{{3}/{2}}} \]

✓ Mathematica. Time used: 0.112 (sec). Leaf size: 48

ode=16*x^2*D[y[x],{x,2}]+32*x*D[y[x],x]+(x^4-12)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {e^{-\frac {1}{2}-\frac {i x^2}{8}} \left (c_1-2 i c_2 e^{1+\frac {i x^2}{4}}\right )}{x^{3/2}} \]

✓ Sympy. Time used: 0.240 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(16*x**2*Derivative(y(x), (x, 2)) + 32*x*Derivative(y(x), x) + (x**4 - 12)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1} J_{\frac {1}{2}}\left (\frac {x^{2}}{8}\right ) + C_{2} Y_{\frac {1}{2}}\left (\frac {x^{2}}{8}\right )}{\sqrt {x}} \]