74.15.52 problem 53 (e)

Internal problem ID [16412]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.7, page 195
Problem number : 53 (e)
Date solved : Thursday, March 13, 2025 at 08:13:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 13

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

\[ y = \frac {x}{x^{2}+1} \]

✓ Mathematica. Time used: 0.426 (sec). Leaf size: 133

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

\[ y(x)\to \frac {\exp \left (\int _1^x\frac {K[1]+2 i}{K[1]^2+1}dK[1]+\int _1^0\frac {K[1]+2 i}{K[1]^2+1}dK[1]\right ) \left (\int _1^x\exp \left (-2 \int _1^{K[2]}\frac {K[1]+2 i}{K[1]^2+1}dK[1]\right )dK[2]-\int _1^0\exp \left (-2 \int _1^{K[2]}\frac {K[1]+2 i}{K[1]^2+1}dK[1]\right )dK[2]\right )}{\sqrt {x^2+1}} \]

✗ Sympy

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

False