83.41.13 problem 4

Internal problem ID [19414]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise at end of chapter VIII. Page 141
Problem number : 4
Date solved : Thursday, March 13, 2025 at 02:24:30 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 30

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

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

✓ Mathematica. Time used: 0.183 (sec). Leaf size: 50

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

\[ y(x)\to x j_1(x) \left (-\left (x^2-3\right ) \cos (x)+3 x \sin (x)+c_1\right )-x y_1(x) \left (\left (x^2-3\right ) \sin (x)+3 x \cos (x)+c_2\right ) \]

✗ Sympy

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

NotImplementedError : solve: Cannot solve -x**2 + (1 - 2/x**2)*y(x) + Derivative(y(x), (x, 2))