75.24.5 problem 745

Internal problem ID [17217]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 18.2. Expanding a solution in generalized power series. Bessels equation. Exercises page 177
Problem number : 745
Date solved : Tuesday, January 28, 2025 at 09:57:49 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.216 (sec). Leaf size: 17

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

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

✓ Solution by Mathematica

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

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

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