59.1.115 problem 117

Internal problem ID [9287]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 117
Date solved : Monday, January 27, 2025 at 06:01:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 6 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (50 x^{2}+1\right ) y^{\prime }+\left (30 x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.035 (sec). Leaf size: 24

dsolve(6*x^2*(1+2*x^2)*diff(y(x),x$2)+x*(1+50*x^2)*diff(y(x),x)+(1+30*x^2)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.277 (sec). Leaf size: 58

DSolve[6*x^2*(1+2*x^2)*D[y[x],{x,2}]+x*(1+50*x^2)*D[y[x],x]+(1+30*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x^{5/12} \left (6 c_2 \sqrt [6]{x}+c_1\right ) \exp \left (-\frac {1}{2} \int _1^x\frac {50 K[1]^2+1}{12 K[1]^3+6 K[1]}dK[1]\right ) \]