59.1.579 problem 595

Internal problem ID [9751]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 595
Date solved : Monday, January 27, 2025 at 06:13:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 16 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+8 x \left (9 x^{2}+1\right ) y^{\prime }+\left (49 x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 21

dsolve(16*x^2*(1+x^2)*diff(y(x),x$2)+8*x*(1+9*x^2)*diff(y(x),x)+(1+49*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {x^{{1}/{4}} \left (c_{2} \ln \left (x \right )+c_{1} \right )}{x^{2}+1} \]

✓ Solution by Mathematica

Time used: 0.241 (sec). Leaf size: 53

DSolve[16*x^2*(1+x^2)*D[y[x],{x,2}]+8*x*(1+9*x^2)*D[y[x],x]+(1+49*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \sqrt {x} (c_2 \log (x)+c_1) \exp \left (-\frac {1}{2} \int _1^x\frac {9 K[1]^2+1}{2 \left (K[1]^3+K[1]\right )}dK[1]\right ) \]