59.1.106 problem 108

Internal problem ID [9278]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 108
Date solved : Monday, January 27, 2025 at 06:00:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+5 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-40 x^{2}+2\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.085 (sec). Leaf size: 35

dsolve(2*x^2*(1+2*x^2)*diff(y(x),x$2)+5*x*(1+6*x^2)*diff(y(x),x)-(2-40*x^2)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.295 (sec). Leaf size: 118

DSolve[2*x^2*(1+2*x^2)*D[y[x],{x,2}]+5*x*(1+6*x^2)*D[y[x],x]-(2-40*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {10 K[1]^2+7}{8 K[1]^3+4 K[1]}dK[1]-\frac {1}{2} \int _1^x\frac {30 K[2]^2+5}{4 K[2]^3+2 K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {10 K[1]^2+7}{8 K[1]^3+4 K[1]}dK[1]\right )dK[3]+c_1\right ) \]