74.15.50 problem 53 (c)

Internal problem ID [16489]
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 (c)
Date solved : Tuesday, January 28, 2025 at 09:09:19 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*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 24

dsolve((1+x^2)^2*diff(y(x),x$2)+2*x*(1+x^2)*diff(y(x),x)+4*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 2.509 (sec). Leaf size: 79

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

\[ y(x)\to \frac {\exp \left (\int _1^x\frac {K[1]+2 i}{K[1]^2+1}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {K[1]+2 i}{K[1]^2+1}dK[1]\right )dK[2]+c_1\right )}{\sqrt {x^2+1}} \]