23.3.251 problem 253

Internal problem ID [5965]
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 : 253
Date solved : Tuesday, September 30, 2025 at 02:07:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 30

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

\[ y = \frac {\left (c_2 x +c_1 \right ) \cos \left (x \right )+\left (c_1 x -c_2 \right ) \sin \left (x \right )+x^{3}}{x} \]

✓ Mathematica. Time used: 0.125 (sec). Leaf size: 50

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

\begin{align*} y(x)&\to x j_1(x) \left (-\left (x^2-3\right ) \cos (x)+3 x \sin (x)+c_1\right )-x y_1(x) \left (\left (x^2-3\right ) \sin (x)+3 x \cos (x)+c_2\right ) \end{align*}

✗ Sympy

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

NotImplementedError : solve: Cannot solve -x**4 + x**2*Derivative(y(x), (x, 2)) + (x**2 - 2)*y(x)