59.1.188 problem 190

Internal problem ID [9360]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 190
Date solved : Monday, January 27, 2025 at 06:01:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+8 x y^{\prime }-\left (-x^{2}+35\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 42

dsolve(4*x^2*(1+x^2)*diff(y(x),x$2)+8*x*diff(y(x),x)-(35-x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {\left (x^{2}+1\right )^{2} c_{2} \ln \left (x^{2}+1\right )+\left (2 x^{2}+\frac {3}{2}\right ) c_{2} +c_{1} \left (x^{2}+1\right )^{2}}{x^{{7}/{2}}} \]

✓ Solution by Mathematica

Time used: 0.174 (sec). Leaf size: 101

DSolve[4*x^2*(1+x^2)*D[y[x],{x,2}]+8*x*D[y[x],x]-(35-x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {K[1]^2-5}{2 \left (K[1]^3+K[1]\right )}dK[1]-\frac {1}{2} \int _1^x\frac {2}{K[2]^3+K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {K[1]^2-5}{2 \left (K[1]^3+K[1]\right )}dK[1]\right )dK[3]+c_1\right ) \]