59.1.754 problem 776

Internal problem ID [9926]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 776
Date solved : Wednesday, March 05, 2025 at 08:01:01 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{4} y^{\prime \prime }+\lambda y&=0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 31

ode:=x^4*diff(diff(y(x),x),x)+lambda*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x \left (c_{1} \sinh \left (\frac {\sqrt {-\lambda }}{x}\right )+c_{2} \cosh \left (\frac {\sqrt {-\lambda }}{x}\right )\right ) \]

✓ Mathematica. Time used: 0.108 (sec). Leaf size: 56

ode=x^4*D[y[x],{x,2}]+\[Lambda]*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 x e^{-1+\frac {i \sqrt {\lambda }}{x}}-\frac {i c_2 x e^{1-\frac {i \sqrt {\lambda }}{x}}}{2 \sqrt {\lambda }} \]

✓ Sympy. Time used: 0.094 (sec). Leaf size: 58

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

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