59.1.521 problem 537

Internal problem ID [9693]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 537
Date solved : Monday, January 27, 2025 at 06:13:10 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x \left (x^{2}+3\right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }-8 y x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 32

dsolve(x*(3+x^2)*diff(y(x),x$2)+(2-x^2)*diff(y(x),x)-8*x*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} \left (x^{2}+3\right )^{{11}/{6}} x^{{1}/{3}}+\frac {c_{2} \left (8 x^{4}+44 x^{2}+55\right )}{8} \]

✓ Solution by Mathematica

Time used: 0.219 (sec). Leaf size: 116

DSolve[x*(3+x^2)*D[y[x],{x,2}]+(2-x^2)*D[y[x],x]-8*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {7 K[1]^2+4}{2 K[1]^3+6 K[1]}dK[1]-\frac {1}{2} \int _1^x\frac {2-K[2]^2}{K[2]^3+3 K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {7 K[1]^2+4}{2 K[1]^3+6 K[1]}dK[1]\right )dK[3]+c_1\right ) \]