59.1.69 problem 71
Internal
problem
ID
[9241]
Book
:
Collection
of
Kovacic
problems
Section
:
section
1
Problem
number
:
71
Date
solved
:
Monday, January 27, 2025 at 06:00:25 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left (3 x^{2}+x +1\right ) y^{\prime \prime }+\left (2+15 x \right ) y^{\prime }+12 y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.220 (sec). Leaf size: 163
dsolve((1+x+3*x^2)*diff(y(x),x$2)+(2+15*x)*diff(y(x),x)+12*y(x)=0,y(x), singsol=all)
\[
y = \frac {{\mathrm e}^{\frac {\sqrt {11}\, \arctan \left (\frac {\left (6 x +1\right ) \sqrt {11}}{11}\right )}{22}} \left (\left (-36 x^{2}-12 x -12\right )^{-\frac {1}{4}+\frac {i \sqrt {11}}{44}} \left (i \sqrt {11}-6 x -1\right )^{{3}/{2}} c_{1} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i \sqrt {11}}{22}, \frac {1}{2}+\frac {i \sqrt {11}}{22}\right ], \left [-\frac {1}{2}+\frac {i \sqrt {11}}{22}\right ], \frac {1}{2}+\frac {i \left (-6 x -1\right ) \sqrt {11}}{22}\right )+\left (i \sqrt {11}-6 x -1\right )^{\frac {5}{4}+\frac {i \sqrt {11}}{44}} \left (i \sqrt {11}+6 x +1\right )^{\frac {5}{4}-\frac {i \sqrt {11}}{44}} \operatorname {hypergeom}\left (\left [2, 2\right ], \left [\frac {5}{2}-\frac {i \sqrt {11}}{22}\right ], \frac {1}{2}+\frac {i \left (-6 x -1\right ) \sqrt {11}}{22}\right ) c_{2} \right )}{\left (3 x^{2}+x +1\right )^{{5}/{4}}}
\]
✓ Solution by Mathematica
Time used: 1.147 (sec). Leaf size: 113
DSolve[(1+x+3*x^2)*D[y[x],{x,2}]+(2+15*x)*D[y[x],x]+12*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to x \exp \left (\int _1^x-\frac {3 K[1]}{6 K[1]^2+2 K[1]+2}dK[1]-\frac {1}{2} \int _1^x\frac {15 K[2]+2}{3 K[2]^2+K[2]+1}dK[2]\right ) \left (c_2 \int _1^x\frac {\exp \left (-2 \int _1^{K[3]}-\frac {3 K[1]}{6 K[1]^2+2 K[1]+2}dK[1]\right )}{K[3]^2}dK[3]+c_1\right )
\]