68.1.2 problem Problem 1.1(b)

Internal problem ID [14131]
Book : Differential Equations, Linear, Nonlinear, Ordinary, Partial. A.C. King, J.Billingham, S.R.Otto. Cambridge Univ. Press 2003
Section : Chapter 1 VARIABLE COEFFICIENT, SECOND ORDER DIFFERENTIAL EQUATIONS. Problems page 28
Problem number : Problem 1.1(b)
Date solved : Tuesday, January 28, 2025 at 06:15:27 AM
CAS classification : [_Lienard]

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

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {\sin \left (x \right )}{x} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 17

dsolve([x*diff(y(x),x$2)+2*diff(y(x),x)+x*y(x)=0,sin(x)/x],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.042 (sec). Leaf size: 37

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

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