76.12.24 problem 36

Internal problem ID [17580]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.2 (Theory of second order linear homogeneous equations). Problems at page 226
Problem number : 36
Date solved : Tuesday, January 28, 2025 at 10:44:28 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 {\sin \left (x \right )}{\sqrt {x}} \end{align*}

✓ Solution by Maple

Time used: 0.215 (sec). Leaf size: 17

dsolve([x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-25/100)*y(x)=0,x^(-1/2)*sin(x)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.037 (sec). Leaf size: 39

DSolve[x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-25/100)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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