59.1.77 problem 79

Internal problem ID [9249]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 79
Date solved : Monday, January 27, 2025 at 06:00:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Solution by Maple

Time used: 0.006 (sec). Leaf size: 16

dsolve((2+4*x)*diff(y(x),x$2)-4*diff(y(x),x)-(6+4*x)*y(x)=0,y(x), singsol=all)
 
\[ y = c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{x} x \]

Solution by Mathematica

Time used: 0.345 (sec). Leaf size: 69

DSolve[(2+4*x)*D[y[x],{x,2}]-4*D[y[x],x]-(6+4*x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 
\[ y(x)\to \sqrt {2 x+1} \exp \left (\int _1^x\left (\frac {1}{-2 K[1]-1}-1\right )dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\left (\frac {1}{-2 K[1]-1}-1\right )dK[1]\right )dK[2]+c_1\right ) \]