59.1.47 problem 49

Internal problem ID [9219]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 49
Date solved : Monday, January 27, 2025 at 05:52:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (2 x +1\right ) y^{\prime }-\left (4+6 x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 46

dsolve(x^2*(1+x)*diff(y(x),x$2)+x*(1+2*x)*diff(y(x),x)-(4+6*x)*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} x^{2}+\frac {c_{2} \left (12 \ln \left (x \right ) x^{4}-12 \ln \left (x +1\right ) x^{4}+12 x^{3}-6 x^{2}+4 x -3\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.253 (sec). Leaf size: 103

DSolve[x^2*(1+x)*D[y[x],{x,2}]+x*(1+2*x)*D[y[x],x]-(4+6*x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {6 K[1]+5}{2 K[1]^2+2 K[1]}dK[1]-\frac {1}{2} \int _1^x\left (\frac {1}{K[2]+1}+\frac {1}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {6 K[1]+5}{2 K[1]^2+2 K[1]}dK[1]\right )dK[3]+c_1\right ) \]