59.1.168 problem 170

Internal problem ID [9340]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 170
Date solved : Monday, January 27, 2025 at 06:01:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.022 (sec). Leaf size: 28

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

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

✓ Solution by Mathematica

Time used: 0.252 (sec). Leaf size: 96

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

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