74.15.52 problem 53 (e)

Internal problem ID [16491]
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 : Tuesday, January 28, 2025 at 09:09:23 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*}

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

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

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