59.1.37 problem 39

Internal problem ID [9209]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 39
Date solved : Monday, January 27, 2025 at 05:52:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 17

dsolve(4*x^2*diff(y(x),x$2)-4*x*diff(y(x),x)+(4*x^2+3)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 39

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

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