59.1.190 problem 192

Internal problem ID [9362]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 192
Date solved : Monday, January 27, 2025 at 06:01:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+5\right ) y^{\prime }-21 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 41

dsolve(x^2*(1+x^2)*diff(y(x),x$2)+x*(5+2*x^2)*diff(y(x),x)-21*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_{1} \left (x^{2}+1\right )^{{5}/{2}} \left (x^{2}+8\right )+35 c_{2} \left (x^{6}+4 x^{4}+\frac {24}{5} x^{2}+\frac {64}{35}\right )}{x^{7}} \]

✓ Solution by Mathematica

Time used: 0.610 (sec). Leaf size: 126

DSolve[x^2*(1+x^2)*D[y[x],{x,2}]+x*(5+2*x^2)*D[y[x],x]-21*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (x^2+8\right ) \exp \left (\int _1^x-\frac {2 K[1]^2+9}{2 \left (K[1]^3+K[1]\right )}dK[1]-\frac {1}{2} \int _1^x\frac {2 K[2]^2+5}{K[2]^3+K[2]}dK[2]\right ) \left (c_2 \int _1^x\frac {\exp \left (-2 \int _1^{K[3]}-\frac {2 K[1]^2+9}{2 \left (K[1]^3+K[1]\right )}dK[1]\right )}{\left (K[3]^2+8\right )^2}dK[3]+c_1\right ) \]