59.1.300 problem 303

Internal problem ID [9472]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 303
Date solved : Monday, January 27, 2025 at 06:03:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 17

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

\[ y = c_{1} \left (x -1\right )+c_{2} {\mathrm e}^{x} x^{2} \]

✓ Solution by Mathematica

Time used: 0.338 (sec). Leaf size: 126

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

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