59.1.169 problem 171

Internal problem ID [9341]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 171
Date solved : Monday, January 27, 2025 at 06:01:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (2+3 x \right ) y^{\prime \prime }+x \left (4+21 x \right ) y^{\prime }-\left (1-9 x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 48

dsolve(2*x^2*(2+3*x)*diff(y(x),x$2)+x*(4+21*x)*diff(y(x),x)-(1-9*x)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.312 (sec). Leaf size: 102

DSolve[2*x^2*(2+3*x)*D[y[x],{x,2}]+x*(4+21*x)*D[y[x],x]-(1-9*x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\left (\frac {1}{K[1]}-\frac {3}{12 K[1]+8}\right )dK[1]-\frac {1}{2} \int _1^x\left (\frac {15}{6 K[2]+4}+\frac {1}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {9 K[1]+8}{12 K[1]^2+8 K[1]}dK[1]\right )dK[3]+c_1\right ) \]