7.9.22 problem 44

Internal problem ID [270]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 2. Linear Equations of Higher Order. Section 2.2 (General solutions of linear equations). Problems at page 122
Problem number : 44
Date solved : Monday, January 27, 2025 at 02:43:09 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*}

Using reduction of order method given that one solution is

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 17

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

\[ y = \frac {c_1 \sin \left (x \right )+c_2 \cos \left (x \right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.044 (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]
 

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