59.1.572 problem 588

Internal problem ID [9744]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 588
Date solved : Monday, January 27, 2025 at 06:13:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (9 x^{2}+1\right ) y^{\prime }+\left (25 x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 41

dsolve(x^2*(1+x^2)*diff(y(x),x$2)-x*(1+9*x^2)*diff(y(x),x)+(1+25*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = x \left (c_{2} \left (x^{4}-4 x^{2}+1\right ) \ln \left (x \right )+c_{1} x^{4}+\left (-4 c_{1} -6 c_{2} \right ) x^{2}+c_{1} +3 c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.609 (sec). Leaf size: 138

DSolve[x^2*(1+x^2)*D[y[x],{x,2}]-x*(1+9*x^2)*D[y[x],x]+(1+25*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (x^4-4 x^2+1\right ) \exp \left (\int _1^x\frac {1-7 K[1]^2}{2 \left (K[1]^3+K[1]\right )}dK[1]-\frac {1}{2} \int _1^x-\frac {9 K[2]^2+1}{K[2]^3+K[2]}dK[2]\right ) \left (c_2 \int _1^x\frac {\exp \left (-2 \int _1^{K[3]}\frac {1-7 K[1]^2}{2 \left (K[1]^3+K[1]\right )}dK[1]\right )}{\left (K[3]^4-4 K[3]^2+1\right )^2}dK[3]+c_1\right ) \]