59.1.136 problem 138

Internal problem ID [9308]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 138
Date solved : Monday, January 27, 2025 at 06:01:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (2 x +1\right ) y^{\prime \prime }+x \left (5+9 x \right ) y^{\prime }+\left (4+3 x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 73

dsolve(x^2*(1+2*x)*diff(y(x),x$2)+x*(5+9*x)*diff(y(x),x)+(4+3*x)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_{2} \left (x +\frac {1}{2}\right )^{2} \ln \left (\sqrt {2 x +1}-1\right )-c_{2} \left (x +\frac {1}{2}\right )^{2} \ln \left (\sqrt {2 x +1}+1\right )+c_{2} \left (x +\frac {2}{3}\right ) \sqrt {2 x +1}+4 \left (x +\frac {1}{2}\right )^{2} c_{1}}{x^{2} \sqrt {2 x +1}} \]

✓ Solution by Mathematica

Time used: 0.216 (sec). Leaf size: 110

DSolve[x^2*(1+2*x)*D[y[x],{x,2}]+x*(5+9*x)*D[y[x],x]+(4+3*x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {7 K[1]+1}{4 K[1]^2+2 K[1]}dK[1]-\frac {1}{2} \int _1^x\frac {9 K[2]+5}{2 K[2]^2+K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {7 K[1]+1}{4 K[1]^2+2 K[1]}dK[1]\right )dK[3]+c_1\right ) \]