59.1.140 problem 142

Internal problem ID [9312]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 142
Date solved : Monday, January 27, 2025 at 06:01:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-3 x \left (-x^{2}+1\right ) y^{\prime }+4 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 27

dsolve(x^2*(1+x^2)*diff(y(x),x$2)-3*x*(1-x^2)*diff(y(x),x)+4*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {x^{2} \left (c_{1} +c_{2} \left (\frac {x^{2}}{2}+\ln \left (x \right )\right )\right )}{\left (x^{2}+1\right )^{2}} \]

✓ Solution by Mathematica

Time used: 0.236 (sec). Leaf size: 107

DSolve[x^2*(1+x^2)*D[y[x],{x,2}]-3*x*(1-x^2)*D[y[x],x]+4*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x-\frac {K[1]^2-1}{2 \left (K[1]^3+K[1]\right )}dK[1]-\frac {1}{2} \int _1^x\frac {3 \left (K[2]^2-1\right )}{K[2]^3+K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}-\frac {K[1]^2-1}{2 \left (K[1]^3+K[1]\right )}dK[1]\right )dK[3]+c_1\right ) \]