73.9.23 problem 14.2 (m)

Internal problem ID [15284]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 14. Higher order equations and the reduction of order method. Additional exercises page 277
Problem number : 14.2 (m)
Date solved : Tuesday, January 28, 2025 at 07:51:33 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {x}{2}-\frac {1}{2 x} \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 15

dsolve([x^2*diff(y(x),x$2)+x*diff(y(x),x)+y(x)=0,sinh(ln(x))],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.016 (sec). Leaf size: 18

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

\[ y(x)\to c_1 \cos (\log (x))+c_2 \sin (\log (x)) \]