59.1.575 problem 591

Internal problem ID [9747]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 591
Date solved : Monday, January 27, 2025 at 06:13:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+2 x \left (-x^{2}+4\right ) y^{\prime }+\left (7 x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.103 (sec). Leaf size: 24

dsolve(4*x^2*diff(y(x),x$2)+2*x*(4-x^2)*diff(y(x),x)+(1+7*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {\left (x^{4}-16 x^{2}+32\right ) \left (c_{1} +2 c_{2} \right )}{32 \sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.400 (sec). Leaf size: 70

DSolve[4*x^2*D[y[x],{x,2}]+2*x*(4-x^2)*D[y[x],x]+(1+7*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {\sqrt {e} \left (x^4-16 x^2+32\right ) \left (c_2 \int _1^x\frac {e^{\frac {K[1]^2}{4}-1}}{K[1] \left (K[1]^4-16 K[1]^2+32\right )^2}dK[1]+c_1\right )}{\sqrt {x}} \]