59.1.549 problem 565

Internal problem ID [9721]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 565
Date solved : Monday, January 27, 2025 at 06:13:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 16 x^{2} y^{\prime \prime }+4 x \left (2 x^{2}+x +6\right ) y^{\prime }+\left (18 x^{2}+5 x +1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.421 (sec). Leaf size: 32

dsolve(16*x^2*diff(y(x),x$2)+4*x*(6+x+2*x^2)*diff(y(x),x)+(1+5*x+18*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-\frac {x \left (x +1\right )}{4}} \left (\left (\int \frac {{\mathrm e}^{\frac {x \left (x +1\right )}{4}}}{x}d x \right ) c_{2} +c_{1} \right )}{x^{{1}/{4}}} \]

✓ Solution by Mathematica

Time used: 0.413 (sec). Leaf size: 57

DSolve[16*x^2*D[y[x],{x,2}]+4*x*(6+x+2*x^2)*D[y[x],x]+(1+5*x+18*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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