59.1.584 problem 600

Internal problem ID [9756]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 600
Date solved : Monday, January 27, 2025 at 06:13:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (x +2\right ) y^{\prime \prime }-x \left (4-7 x \right ) y^{\prime }-\left (5-3 x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.044 (sec). Leaf size: 55

dsolve(2*x^2*(2+x)*diff(y(x),x$2)-x*(4-7*x)*diff(y(x),x)-(5-3*x)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {15 \,\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {x +2}}{2}\right ) c_{2} x^{3}+33 c_{2} \sqrt {2}\, \left (x^{2}+\frac {52}{33} x +\frac {32}{33}\right ) \sqrt {x +2}+c_{1} x^{3}}{\sqrt {x}\, \left (x +2\right )^{{7}/{2}}} \]

✓ Solution by Mathematica

Time used: 0.257 (sec). Leaf size: 106

DSolve[2*x^2*(2+x)*D[y[x],{x,2}]-x*(4-7*x)*D[y[x],x]-(5-3*x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\left (\frac {2}{K[1]}-\frac {5}{4 (K[1]+2)}\right )dK[1]-\frac {1}{2} \int _1^x\left (\frac {9}{2 (K[2]+2)}-\frac {1}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {3 K[1]+16}{4 K[1]^2+8 K[1]}dK[1]\right )dK[3]+c_1\right ) \]