59.1.661 problem 678

Internal problem ID [9833]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 678
Date solved : Monday, January 27, 2025 at 06:14:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-\left (2 \sqrt {5}-1\right ) x y^{\prime }+\left (\frac {19}{4}-3 x^{2}\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 29

dsolve(x^2*diff(y(x),x$2)-(2*sqrt(5)-1)*x*diff(y(x),x)+(19/4-3*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = x^{-\frac {1}{2}+\sqrt {5}} \left (c_{1} \sinh \left (\sqrt {3}\, x \right )+c_{2} \cosh \left (\sqrt {3}\, x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.076 (sec). Leaf size: 53

DSolve[x^2*D[y[x],{x,2}]-(2*Sqrt[5]-1)*x*D[y[x],x]+(19/4-3*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{6} e^{-\sqrt {3} x} x^{\sqrt {5}-\frac {1}{2}} \left (\sqrt {3} c_2 e^{2 \sqrt {3} x}+6 c_1\right ) \]