59.1.154 problem 156

Internal problem ID [9326]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 156
Date solved : Monday, January 27, 2025 at 06:01:31 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 48

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

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

✓ Solution by Mathematica

Time used: 0.225 (sec). Leaf size: 112

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

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