59.1.167 problem 169

Internal problem ID [9339]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 169
Date solved : Monday, January 27, 2025 at 06:01:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (-2 x^{2}+1\right ) y^{\prime \prime }+x \left (-13 x^{2}+7\right ) y^{\prime }-14 x^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 35

dsolve(x^2*(1-2*x^2)*diff(y(x),x$2)+x*(7-13*x^2)*diff(y(x),x)-14*x^2*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_{1} \left (2 x^{2}-1\right )^{{5}/{4}}+5 c_{2} x^{4}-20 c_{2} x^{2}+8 c_{2}}{x^{6}} \]

✓ Solution by Mathematica

Time used: 0.236 (sec). Leaf size: 116

DSolve[x^2*(1-2*x^2)*D[y[x],{x,2}]+x*(7-13*x^2)*D[y[x],x]-14*x^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {5-K[1]^2}{4 K[1]^3-2 K[1]}dK[1]-\frac {1}{2} \int _1^x\frac {7-13 K[2]^2}{K[2]-2 K[2]^3}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {5-K[1]^2}{4 K[1]^3-2 K[1]}dK[1]\right )dK[3]+c_1\right ) \]