59.1.74 problem 76

Internal problem ID [9246]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 76
Date solved : Monday, January 27, 2025 at 06:00:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}+x +1\right ) y^{\prime \prime }+\left (1+7 x \right ) y^{\prime }+2 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.064 (sec). Leaf size: 77

dsolve((1+x+2*x^2)*diff(y(x),x$2)+(1+7*x)*diff(y(x),x)+2*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} \operatorname {hypergeom}\left (\left [\frac {1}{2}, 2\right ], \left [-\frac {\left (-7 \sqrt {7}+3 i\right ) \sqrt {7}}{28}\right ], \frac {1}{2}+\frac {i \left (-4 x -1\right ) \sqrt {7}}{14}\right )+c_{2} \left (i \sqrt {7}+4 x +1\right )^{-\frac {3}{4}+\frac {3 i \sqrt {7}}{28}} \left (i \sqrt {7}-4 x -1\right )^{-\frac {3}{4}-\frac {3 i \sqrt {7}}{28}} \left (x +1\right ) \]

✓ Solution by Mathematica

Time used: 1.008 (sec). Leaf size: 119

DSolve[(1+x+2*x^2)*D[y[x],{x,2}]+(1+7*x)*D[y[x],x]+2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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