59.1.532 problem 548

Internal problem ID [9704]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 548
Date solved : Monday, January 27, 2025 at 06:13:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 34

dsolve(4*x^2*(1+x^2)*diff(y(x),x$2)+4*x*(1+6*x^2)*diff(y(x),x)-(1-25*x^2)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.392 (sec). Leaf size: 70

DSolve[4*x^2*(1+x^2)*D[y[x],{x,2}]+4*x*(1+6*x^2)*D[y[x],x]-(1-25*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {\left (c_2 x \text {arcsinh}(x)-c_2 \sqrt {x^2+1}+c_1 x\right ) \exp \left (-\frac {1}{2} \int _1^x\frac {6 K[1]^2+1}{K[1]^3+K[1]}dK[1]\right )}{\sqrt [4]{x^2+1}} \]