59.1.59 problem 61

Internal problem ID [9231]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 61
Date solved : Monday, January 27, 2025 at 05:52:21 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}-8 x +11\right ) y^{\prime \prime }-16 \left (x -2\right ) y^{\prime }+36 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 55

dsolve((11-8*x+2*x^2)*diff(y(x),x$2)-16*(x-2)*diff(y(x),x)+36*y(x)=0,y(x), singsol=all)
 

\[ y = c_{2} x^{6}-12 c_{2} x^{5}+\frac {165 c_{2} x^{4}}{2}+c_{1} x^{3}+\frac {3 \left (-8 c_{1} -1815 c_{2} \right ) x^{2}}{4}+\frac {3 \left (37 c_{1} +10890 c_{2} \right ) x}{10}-\frac {31 c_{1}}{5}-\frac {16577 c_{2}}{8} \]

✓ Solution by Mathematica

Time used: 0.625 (sec). Leaf size: 146

DSolve[(11-8*x+2*x^2)*D[y[x],{x,2}]-16*(x-2)*D[y[x],x]+36*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} \left (2 x+5 i \sqrt {6}-4\right ) \left (2 x^2-8 x+11\right )^2 \exp \left (\int _1^x\frac {2 K[1]+5 i \sqrt {6}-4}{2 (K[1]-4) K[1]+11}dK[1]\right ) \left (c_2 \int _1^x-\frac {4 \exp \left (-2 \int _1^{K[2]}\frac {2 K[1]+5 i \sqrt {6}-4}{2 (K[1]-4) K[1]+11}dK[1]\right )}{\left (-2 i K[2]+5 \sqrt {6}+4 i\right )^2}dK[2]+c_1\right ) \]