75.24.8 problem 748

Internal problem ID [17149]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 18.2. Expanding a solution in generalized power series. Bessels equation. Exercises page 177
Problem number : 748
Date solved : Monday, March 31, 2025 at 03:43:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-2 x y^{\prime }+4 \left (x^{4}-1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 45

ode:=x^2*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+4*(x^4-1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {-\frac {\operatorname {BesselY}\left (\frac {1}{4}, x^{2}\right ) c_2}{2}-\frac {\operatorname {BesselJ}\left (\frac {1}{4}, x^{2}\right ) c_1}{2}+x^{2} \left (\operatorname {BesselJ}\left (-\frac {3}{4}, x^{2}\right ) c_1 +\operatorname {BesselY}\left (-\frac {3}{4}, x^{2}\right ) c_2 \right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.144 (sec). Leaf size: 46

ode=x^2*D[y[x],{x,2}]-2*x*D[y[x],x]+4*(x^4-1)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {x^{3/2} \left (c_2 \operatorname {Gamma}\left (\frac {9}{4}\right ) \operatorname {BesselJ}\left (\frac {5}{4},x^2\right )-4 c_1 \operatorname {Gamma}\left (\frac {3}{4}\right ) \operatorname {BesselJ}\left (-\frac {5}{4},x^2\right )\right )}{2^{3/4}} \]

✓ Sympy. Time used: 0.223 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = x^{\frac {3}{2}} \left (C_{1} J_{\frac {5}{4}}\left (x^{2}\right ) + C_{2} Y_{\frac {5}{4}}\left (x^{2}\right )\right ) \]