23.3.528 problem 534

Internal problem ID [6242]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 534
Date solved : Tuesday, September 30, 2025 at 02:38:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 30

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

\[ y = \left (\left (c_1 x +c_2 \right ) \cos \left (\frac {1}{x}\right )-\sin \left (\frac {1}{x}\right ) \left (c_2 x -c_1 \right )\right ) x \]

✓ Mathematica. Time used: 0.117 (sec). Leaf size: 47

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

\begin{align*} y(x)&\to c_1 e^{i/x} x (x-i)+\frac {1}{2} c_2 e^{-i/x} (1-i x) x \end{align*}

✓ Sympy. Time used: 0.056 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 2)) + (1 - 2*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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