59.1.319 problem 326

Internal problem ID [9491]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 326
Date solved : Monday, January 27, 2025 at 06:03:24 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 2 x y^{\prime \prime }-y^{\prime }+2 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 36

dsolve(2*x*diff(y(x),x$2)-diff(y(x),x)+2*y(x)=0,y(x), singsol=all)
 

\[ y = \left (2 \sqrt {x}\, c_{1} +c_{2} \right ) \cos \left (2 \sqrt {x}\right )-\sin \left (2 \sqrt {x}\right ) \left (-2 c_{2} \sqrt {x}+c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.205 (sec). Leaf size: 74

DSolve[2*x*D[y[x],{x,2}]-D[y[x],x]+2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{2 i \sqrt {x}} \left (2 \sqrt {x}+i\right ) \left (c_2 \int _1^x\frac {e^{-4 i \sqrt {K[1]}} \sqrt {K[1]}}{\left (2 \sqrt {K[1]}+i\right )^2}dK[1]+c_1\right ) \]