61.28.1 problem 61

Internal problem ID [12561]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 61
Date solved : Tuesday, January 28, 2025 at 03:21:40 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 27

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

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

✓ Solution by Mathematica

Time used: 0.026 (sec). Leaf size: 38

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

\[ y(x)\to c_1 \cos \left (2 \sqrt {a} \sqrt {x}\right )+c_2 \sin \left (2 \sqrt {a} \sqrt {x}\right ) \]